Multivariate bounded variation functions of Jordan–Wiener type

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Multivariate bounded variation functions of Jordan–Wiener type Alexander Brudnyia ,∗,1 , Yuri Brudnyib a

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4, b Department of Mathematics, Technion, Haifa, 32000, Israel

Received 11 December 2018; received in revised form 15 November 2019; accepted 21 November 2019 Available online 29 November 2019 Communicated by H.G. Feichtinger

Abstract We introduce and study spaces of multivariate functions of bounded variation generalizing the classical Jordan and Wiener spaces. Multivariate generalizations of the Jordan space were given by several prominent researchers. However, each of the proposed concepts preserves only few properties of Jordan variation which are designed to a selected application. In contrast, the multivariate generalization of the Jordan space presented in this paper preserves all known and reveals some previously unknown properties of the space. These, in turn, are special cases of the basic properties of the introduced spaces proved in the paper. Specifically, the first part of the paper describes structure properties of functions of bounded (k, p)-variation (V pk functions). It includes assertions on discontinuity sets and pointwise differentiability of V pk functions and their Luzin type and C ∞ approximations. The second part presents results on Banach structure of V pk spaces, namely, atomic decomposition and constructive characterization of their predual spaces. As a result, we obtain the so-called two-stars theorems describing V pk spaces as second duals of their separable subspaces consisting of functions of “vanishing variation”. c 2019 Elsevier Inc. All rights reserved. ⃝ MSC: primary 26B30; secondary 46E35 Keywords: Spaces of bounded variation; Oscillation; Local polynomial approximation; Packing; Smoothness; Signed Borel measure; Atom; Chain; Duality; Predual space; Two-stars theorem

∗ Corresponding author. 1

E-mail addresses: [email protected] (A. Brudnyi), [email protected] (Y. Brudnyi). Research of the first author is supported in part by NSERC, Canada.

https://doi.org/10.1016/j.jat.2019.105346 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝

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1. Introduction 1.1. Main concept (motivation) The paper is devoted to the study of a new family of multivariate function spaces generalizing the classical Jordan–Wiener spaces {V p }. A generic space of the family is denoted by V˙ pk (Q d ), where k ∈ N, 1 ≤ p < ∞ and Q d is a d-dimensional closed cube; in what follows, without loss of generality we take Q d := [0, 1]d .

(1.1)

The space consists of bounded on Q d functions with bounded (k, p)-variation denoted by | f |V pk . To introduce and motivate this notion we use an equivalent form of the classical p-variation given for a function f ∈ ℓ∞ [0, 1] by ) 1p ( ∑ p ; (1.2) | f |V p := sup osc( f ; I ) π

I ∈π

here π is a packing, i.e., a set of pairwise nonoverlapping2 closed intervals I ⊂ [0, 1] and osc( f ; I ) := sup | f (x) − f (y)|

(1.3)

x,y∈I

is the oscillation of f on I . The reader can easily see that (1.2) is equivalent to the classical Jordan definition with osc( f ; I ) replaced by deviation δ( f ; I ) := | f (b) − f (a)|, where I := [a, b] and π runs over partitions, i.e., coverings of [0, 1] by pairwise nonoverlapping intervals. It seems to be natural to define multivariate spaces of Jordan–Wiener type by taking in (1.2) f ∈ ℓ∞ (Q d ) and π being a family of pairwise nonoverlapping closed subcubes Q ⊂ Q d . However, assuming boundedness of such variation we obtain the family {ℓ p (Q d )}1≤ p<∞ that has no relation to the required function spaces. To overcome this obstacle we enlarge the order of difference in (1.3) using a function of a cube Q ↦→ osck ( f ; Q), Q ⊂ Q d , given by osck ( f ; Q) := sup {|∆kh f (x)| : x + j h ∈ Q; j = 0, . . . , k};

(1.4)

h∈Rd

here as usual ∆kh

( ) k ∑ k− j k f := (−1) f (· + j h). j j=0

(1.5)

The family of spaces V˙ pk (Q d ) obtained in this way is the main object of our study. Remark 1.1. The definition of V˙ pk (Q d ) is meaningful only in the case of k ≥ dp ; otherwise, d V˙ pk (Q d ) = ℓ p (Q d )+Pk−1 , where the second term stands for the space of polynomials of degree d ≤ k − 1 on R , see Theorem 2.1. To illustrate the results of the present paper we briefly discuss their special cases for the space V˙1d (Q d ) that may be regarded as a d-dimensional analog of Jordan space V (= V˙11 (Q 1 ) 2

Two subsets of a topological space are nonoverlapping if their interiors are disjoint.

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in our notation). We justify its role as a “genuine” analog of Jordan’s space for d ≥ 2 by comparing its properties presented below with those for d = 1. In what follows, V˙1d (Q d ) is denoted for brevity by V˙ (Q d ). (a) For every d ≥ 1, functions of V˙ (Q d ) have at most countable sets of discontinuities, see Theorem 2.1. (b) For every d ≥ 1, functions of V˙ (Q d ) are d-differentiable in the Peano sense3 almost everywhere (a.e.). In particular, this implies for d = 1 the classical Lebesgue differentiability theorem, see Theorem 2.15 (a). (c) Given f ∈ V˙ (Q d ), d ≥ 1, and ε ∈ (0, 1) there is a function f ε ∈ C d (Q d ) such that mesd {x ∈ Q d : f (x) ̸= f ε (x)} < ε, see Theorem 2.11. (d) Let AC(Q d ), d ≥ 1, be a closed subspace of V˙ (Q d ) consisting of functions f ∈ ℓ∞ (Q d ) satisfying ∑ osck ( f ; Q) = 0; lim |π|→0

Q∈π

here and in what follows |π | := Then it is true that

∑

Q∈π

|Q| and |S| is the d-measure of S ⊂ Rd .

AC(Q d ) = W˙ 1d (Q d ), where hereafter W˙ pk (Q d ) stands for the homogeneous Sobolev space of order k over L p (Q d ). Moreover, every f ∈ AC(Q d ) is restored by some integral operator up to an additive term being a polynomial of degree ≤ d − 1 via the distributional gradient ∇ d f := (D α f )|α|=d , see Corollary 2.20. For d = 1, this gives the so-called Fundamental Theorem of Calculus by Lebesgue–Vitali. (e) Let N V˙ (Q d ), d ≥ 1, be the normalized part of V˙ (Q d ) so that V˙ (Q d ) = N V˙ (Q d ) ⊕ ℓ1 (Q d ), see Section 2.1. Then it is true that N V˙ (Q d ) = B V˙ d (Q d ), where B V˙ k (Q d ) is the Sobolev space of order k whose distributional kth derivatives are bounded Borel measures, see Theorem 2.18 (b). For d = 1, this gives the classical Hardy–Littlewood theorem [27] as B V˙ 1 (Q 1 ) coincides with the Lipschitz space over L 1 (Q 1 ). (f) If a sequence { f n }n∈N ⊂ V˙ (Q d ), d ≥ 1, is such that sup | f n |V (Q d ) < ∞, n∈N

then for some sequences J ⊂ N of integers and { pn }n∈J of polynomials of degree ≤ d − 1 the sequence { f n + pn }n∈J converges pointwise to a function f ∈ V˙ (Q d ) and, moreover, | f |V (Q d ) ≤ lim | f n |V (Q d ) . n∈J

For d = 1, this gives the classical Helly selection theorem (see, e.g., [38]). 3

See the corresponding definition in Section 2.2.

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The variety of useful properties of the space V˙ (Q d ) (more generally, spaces V˙ pk (Q d )) allows their applicability to various fields of modern analysis including multivariate harmonic analysis, N -term approximation, the real interpolation of smoothness spaces. The corresponding results will be presented in the forthcoming papers. Remark 1.2. There are two more spaces of the family, namely, V˙dd (Q d ) and V˙d1 (Q d ) which for d = 1 coincide with the space V [0, 1]. However, they do not possess almost all of the above formulated properties for V˙ (Q d ) (e.g., if d ≥ 2, then V˙dd (Q d ) contains nowhere differentiable functions). 1.2. Historical remarks The important role of the Jordan space V in univariate function theory has attracted several prominent researchers to the problem of its multivariate generalization. The introduced notions single out multivariate function spaces preserving only some particular properties of the Jordan space. These properties, in turn, were used to solve a specific problem of multivariate analysis. In this paper, we briefly discuss only those works that have been influenced further development of the multivariate theory. (a) Vitali variation (associated with the names of Vitali [47], Lebesgue [32], Vall´ee Poussin [46]). Let [x, y] ⊂ Rd , x = (x1 , . . . , xd ), y = (y1 , . . . , yd ), be a d-interval, i.e., the set [x, y] := {z = (z 1 , . . . , z d ) ∈ Rd : xi ≤ z i ≤ yi , i = 1, . . . , d}. For a function f ∈ ℓ∞ (I ), I := [x, y], we define the deviation by setting δ( f ; I ) :=

1 ∑

···

j1 =0

1 ∑

(−1) j1 +···+ jd f (y1 + j1 (x1 − y1 ), . . . , yd + jd (xd − yd ));

(1.6)

jd =0

in particular, for d = 2 δ( f ; I ) = f (y1 , y2 ) − f (y1 , x2 ) − f (x1 , y2 ) + f (x1 , x2 ). The Vitali variation is given for f ∈ ℓ∞ (Q d ) by { } ∑ |δ( f ; I )| , varv f := sup π

(1.7)

I ∈π

where π runs over all families of pairwise nonoverlapping d-intervals in Q d . The space of functions on Q d with bounded Vitali variations is denoted by Vv (Q d ). Remark 1.3. (a) In the original definition, see, e.g., [1], π runs over all partitions of Q d by such d-intervals. Since ∑ δ( f ; I ) = δ( f ; I ′ ), I ′ ∈π

where π is a partition of I in nonoverlapping d-intervals I ′ by hyperplanes parallel to the coordinate ones, this definition is equivalent to (1.7). (b) The deviations in (1.7) can be clearly replaced by oscillations; here osc( f ; I ) := sup |δ( f ; I ′ )|, I ′ ⊂I

where supremum is taken over all d-subintervals of I .

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Despite the similarity of these definitions the structure of functions from Vv (Q d ) is essentially poorer than that for functions from V˙ (Q d ). In particular, the former contains nonmeasurable functions and nowhere differentiable functions. Nevertheless, the affinity leads to the following continuous embedding V˙ (Q d ) ⊂ Vv (Q d ).

(1.8)

Finally, let us mention the important application of the Vitali variation to the study of properties of Riemann–Stiltjes type multiple integrals, see [40]. (b) Hardy–Krause variation. It is denoted by varh f and defined via the Vitali variation as follows. Let {ei }1≤i≤n be the standard basis of Rd and xˆ ∈ Q d , ω ⊂ {1, . . . , d} be fixed. Then a partial function of f ∈ ℓ∞ (Q d ) defined by x, ˆ ω is given by ⎞ ⎛ ∑ ∑ f x,ω : (xi )i∈ω ↦→ f ⎝ x i ei + xˆi ei ⎠ ; ˆ i∈ω

i̸∈ω

this is clearly defined on the unit cube Q ω ∼ = [0, 1]|ω| of dimension |ω| in the subspace of Rd i generated by vectors e , i ∈ ω. Now Hardy–Krause variation is given for f ∈ ℓ∞ (Q d ) by { } ∑ varh f := sup varv f x,ω ; (1.9) ˆ xˆ

ω

here ω runs over all nonempty subsets of {1, . . . , d}. The space of functions of bounded Hardy–Krause variation is denoted by Vh (Q d ). The functions of this subspace of Vv (Q d ) have essentially better properties. In particular, every f ∈ Vh (Q d ) has one-sided limits in each variable and is differentiable a.e. The former is the matter of definition and the latter is proved in [14] for d = 2 (the proof can be easily extended to all d). The main application of Hardy–Krause variation is a generalization of Dirichlet–Jordan convergence criterion for multiple Fourier series, see [26] for d = 2 and [37] for d > 2. For some modern development of these results in multivariate harmonic analysis, approximation theory and numerical computations see, e.g., [33] and references therein. (c) Tonelli variation. Let∑f ∈ ℓ∞ (Q d ) and f x i : xi ↦→ f (x), 0 ≤ xi ≤ 1, be a univariate partial function with i x := j̸=i x j e j being fixed, 1 ≤ i ≤ d. Tonelli variation for this f is given by d ∫ ∑ vart f := (var f x i ) d x i ; (1.10) i=1

Q d−1

here the (Jordan) variations are assumed to be Lebesgue integrable. This notion is introduced to characterize continuous functions on Q d whose graphs are rectifiable, see [44] for d = 2 and proof in [42, Ch. 5] that is easily extended to d > 2. For integrable functions, Tonelli variation loses its geometric meaning, see [2]. The corresponding adaptation to this case was given by Cesari [17]; the modern version exploiting the theory of distributions is presented in the books [25] and [3]. The corresponding variation for

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f ∈ L 1 (Q d ) is given by var f := Qd

d ∑

var ∂i f,

(1.11)

Qd

i=1

where the distributional partial derivatives are assumed to be finite Borel measures. The corresponding space is denoted by BV (Q d ). This definition is given in essence by De Giorgi [19]; its equivalence to that of Tonelli–Cesari is proved by Krickeberg [30]. In particular, for f ∈ C(Q d ) vart f ≈ var f with constants of equivalence depending only on d. Nevertheless, there is some connection of this space with a space of the family {V˙ pk (Q d )}. Actually, using the equivalent definition of var f given for f ∈ L 1 (Q d ), d ≥ 2, by Qd

var f = sup Qd

where f Q :=

π

1 |Q|

∑ (∫

Q

) d−1 d dx

,

Q

Q∈π

∫

| f − fQ|

d d−1

f d x, see [9, Sec. 1.3.4], and then applying the H¨older inequality we have

var f ≤ c(d)| f |V 1 (Q d ) . d

Qd

Finally, we present one more variation given by Kronrod [31] for d = 2 and by Vitushkin [48] for all d. It is given only for continuous functions on Q d by ∫ varkv f := Hd−1 {x ∈ Q d : f (x) = t} dt, (1.12) R

where Hd−1 is the (d − 1)-Hausdorff measure. For d = 1, this coincides with the Jordan variation of f by the Banach indicatrix theorem and for d > 1 varkv f ≈ vart f with constants of equivalence depending only on d, see [31]. In fact, definitions in the cited papers include also notions of variations related to sections of function graphs by planes of dimensions 1 ≤ s ≤ d −1. For applications of these characteristics to the geometry of sets, approximation theory and the 13th Hilbert problem, see [49]. 1.3. Basic definitions Throughout the rest of the paper we use the following: Notation 1.4. N = {1, 2, . . . }, Z, R denote the sets of natural, integer and real numbers, respectively. Rd is the d-dimensional Euclidean space of vectors x = (x1 , . . . , xd ), y, z, etc. If S ⊂ Rd , then S¯ denotes its closure and S˚ = int S its interior. The d-dimensional Lebesgue measure of S is denoted by |S|. The characteristic function (indicator) of S is denoted by χ S .

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As above, Q d denotes the unit cube [0, 1]d . Q, Q ′ , etc. denote subcubes of Rd homothetic to Q (named further cubes or subcubes); Q r (x) denotes the cube centered at x of sidelength 2r . π, π ′ , πi , etc. denote packings, i.e., families of nonoverlapping cubes Q ⊂ Rd . Π (S) (briefly, Π for S = Q d ) denotes the set of packings in S of finite cardinality contained in S. Pkd denotes the linear space of polynomials in x ∈ Rd of degree ≤ k, i.e., the linear hull (also called linear span) of monomials x α , |α| ≤ k, where d

x α :=

d ∏

α

xi i ,

|α| :=

i=1

d ∑

αi ,

α ∈ Zd+ .

i=1

ℓ p (S), 1 ≤ p ≤ ∞, S ⊂ Rd , denotes the Banach space of functions f : S → R defined by norm { } 1p ∑ ∥ f ; S∥ p := | f (x)| p ; x∈S

in particular, ∥ f ; S∥∞ := sup | f |. S

All f ∈ ℓ p (S) are clearly bounded on S and their supports supp f := {x ∈ S : f (x) ̸= 0} are at most countable for p < ∞. d d ℓloc p (R ) consists of functions on R whose traces to every compact subset K belong to ℓ p (K ). C(S), S ⊂ Rd , denotes the Banach space of bounded continuous functions f : S → R equipped with norm ∥ f ∥C(S) := sup S | f |. C ∞ (S) denotes the linear space of traces f | S of functions f ∈ C ∞ (Rd ). Let X, Y be linear (semi-) normed vector spaces. We write X ↪→ Y

(1.13)

if there is a linear continuous injection of X into Y , and replace ↪→ by ⊂ if X is a linear subspace of Y and the natural embedding operator X → Y is continuous. Further, we say that these spaces are isomorphic and write X∼ =Y

(1.14)

if X ↪→ Y and Y ↪→ X and composition X → X of these continuous injections is the identity map; moreover, we write X =Y if, in addition, they coincide as linear spaces, hence, have equivalent (semi-) norms. Moreover, spaces X and Y are said to be isometrically isomorphic if injections in (1.14) are of norm 1. We write in this case X ≡ Y.

(1.15)

Finally, we denote by B(X ) the closed unit ball of a (semi-) normed space X , i.e., B(X ) := {x ∈ X : ∥x∥ X ≤ 1}.

(1.16)

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Now we present the basic notions of the paper. 1.3.1. Local polynomial approximation In the sequel, we use a more suitable for applications definition of the (k, p)-variation obtained by replacing osck in the initial definition by a function of cube E k ( f ; ·) called local polynomial approximation, see Definition 1.5. The relation between these two set-functions is given for I ⊂ R by the equivalence [50] E k ( f ; I ) ≈ osck ( f ; I ),

(1.17)

where constants of equivalence depend only on k. The general result of this kind proved below, see Theorem 6.1, shows that these two definitions of the (k, p)-variation are equivalent. However, the second definition is more suitable for applications since it relies on various techniques of local approximation theory and so is easier to apply to the study of the Banach structure of V pk (Q d ) spaces and their elements. (For instance, the fact that the (k, p)-variation of f ∈ ℓ∞ (Q d ) is zero iff f is a polynomial of degree ≤ k − 1 trivially follows from the second definition but is highly nontrivial in the framework of the first one.) Definition 1.5.

Local polynomial approximation of order k ∈ N is a function

E k : ( f, S) ↦→ R+ ,

d d f ∈ ℓloc ∞ (R ), S ⊂ R ,

given by E k ( f ; S) :=

inf ∥ f − m; S∥∞ .

(1.18)

d m∈Pk−1

This function possesses several important properties, see, in particular, [7, § 2], that will be used in the forthcoming proofs and mostly are proved there. 1.3.2. (k,p)-variation Using the previous notion we introduce a set-function S ↦→ varkp ( f ; S), k ∈ N, 1 ≤ p < ∞, called (k, p)-variation. d Definition 1.6. The (k, p)-variation of a locally bounded function f ∈ ℓloc ∞ (R ) on a bounded d set S ⊂ R is given by ⎛ ⎞ 1p ∑ (1.19) varkp ( f ; S) := sup ⎝ E k ( f ; Q) p ⎠ ; π ∈Π (S)

Q∈π

this equals 0 if S˚ = ∅. It is the matter of definition to verify the next properties of the object introduced. Proposition 1.7 (Subadditivity). If {Si }i∈N is a sequence of disjoint bounded sets in Rd with ∞ ∪i=1 Si also bounded, then (∞ ) 1p ( ∞ ) ∑ ⋃ k p k var p ( f ; Si ) ≤ var p f ; Si . i=1

i=1

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(Lower semicontinuity) If a sequence { f i }i∈N ⊂ ℓ∞ (S) converges in this space to a function f , then varkp ( f ; S) ≤ lim varkp ( f i ; S). i→∞

(Monotonicity) The function (k, p, S) ↦→ varkp ( f ; S) is nondecreasing in S and nonincreasing in k, p. 1.3.3. V pk spaces Now we consider varkp as a function of f . Using again its definition we have the following: Proposition 1.8. function

Let S ⊂ Rd be the closure of a domain (open connected set). Then the

f ↦→ varkp ( f ; S) ∈ [0, ∞],

f ∈ ℓloc ∞ (S),

d satisfies properties of a seminorm and equals 0 iff f ∈ Pk−1 |S .

Now we present the main object of our study. Definition 1.9. (k, p)-variation

The linear space V˙ pk (Q d ) consists of functions f

∈ ℓ∞ (Q d ) whose

| f |V pk := varkp ( f ; Q d )

(1.20)

is finite. d By Proposition 1.8 (V˙ pk (S), | · |V pk ) is a seminormed space with Pk−1 | Q d being its null-space. k ˙ The standard argument shows that V p (S) is complete.

Stipulation 1.10. Since Q d := [0, 1]d is fixed, throughout the paper we remove it from d d notations writing, e.g., V˙ pk , C ∞ , Pk−1 instead of V˙ pk (Q d ), C ∞ (Q d ), Pk−1 |Qd . In some cases, it will be more appropriate to use a seminorm on V˙ pk defined by replacing d in (1.19) E k ( f ; Q) by osck ( f ; Q), Q ∈ π . The latter is given for f ∈ ℓloc ∞ (R ) and a bounded d subset S ⊂ R by osck ( f ; S) := sup {|∆kh f (x)| : x + j h ∈ S, j = 0, . . . , k},

(1.21)

h∈Rd

cf. definition (1.4). Our work is divided into two parts where the first one studies pointwise behavior of V˙ pk functions while the second is focused on the duality structure of a Banach space V pk associated to V˙ pk . Among different equivalent presentations of this space we prefer to use the quotient of V˙ pk by its null-space, i.e., we define an associated to V˙ pk Banach space by the formula d V pk := V˙ pk /Pk−1

and denote by ∥ · ∥V pk its (factor)-norm. d If f˜ ∈ V pk is a factor-class { f } + Pk−1 , f ∈ V˙ pk , then clearly ∥ f˜∥V pk = | f |V pk . By this reason, we call elements of V pk functions and denote them by f, g, etc.

(1.22)

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In both parts of the work, the essential role plays a numerical characteristic of V˙ pk given by d (1.23) p and called smoothness. The following motivates the name. A function f ∈ V˙ pk satisfies the Lipschitz condition of order s (:= s(V˙ pk )) a.e. if 0 < s < k, has the kth Peano differential a.e. if s = k, and equals a polynomial of degree ≤ k − 1 outside of at most countable subset of Q d if s > k, see Theorem 2.1, 2.15. Moreover, in the second part we essentially use a closed separable subspace4 of V˙ pk denoted by V˙ kp (respectively, Vkp ⊂ V pk ) given by s(V˙ pk ) :=

k

˙p V

:= clos(C ∞ ∩ V˙ pk , V˙ pk ).

(1.24)

Finally, let us single out another multivariate generalization of the Jordan–Wiener (- F. Riesz) concept of variation where local polynomial approximation is taken in the L q (Rd )-metric, 1 ≤ q ≤ ∞, and ℓ p in (1.19) is replaced by a weighted ℓ p . This concept has been studied in our previous paper [9] some results of which for q = ∞ are related to those presented here, see Theorem 2.5. 1.4. Outline of the paper The paper is organized as follows. Section 2, devoted to the formulations of the main results, has four subsections. Section 2.1 contains results describing the structure of V˙ pk function spaces. Section 2.2 presents several results describing relations between V˙ pk and the Lipschitz spaces over ℓ∞ and L p of order s := dp ∈ (0, k]. These include Luzin type approximation of V˙ pk functions by Lipschitz and C k functions, the Lebesgue type theorem on their pointwise differentiability a.e. of order s and the Hardy–Littlewood type theorem on two-sided embeddings connecting V˙ pk with Lipschitz–Besov spaces over L p . As a consequence, the latter gives a multivariate generalization of the Lebesgue–Vitali Fundamental Theorem of Calculus. Section 2.3 introduces a predual to V pk Banach space denoted by U pk and describes its Banach structure and the relation (U pk )∗ ≡ V pk . Section 2.4 contains the so-called two-stars theorem relating the second dual of the space V kp to the nonseparable space V pk . The formulations preceding this result describe measure theoretic properties of some dense subspace of U pk and following from them and the F. Riesz representation theorem the relation (Vkp )∗ ∼ = U pk . The remaining sections contain proofs of the formulated results. 2. Formulation of the main results 2.1. Structure properties of V˙ pk functions Our first result shows that V˙ pk functions initially assumed to be only bounded (not necessarily measurable) have much finer structure. 4

As we will see, space V˙ pk is nonseparable.

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Theorem 2.1. Let f be a V˙ pk function of smoothness s. Then the following is true. (a) If d = 1, then f has one-sided limits at each f ∈ [0, 1]. (b) If d ≥ 2, then f has a limit at each point x ∈ Q d . In both cases, f has at most countable set of discontinuity points. d (c) If s > k, then the vector space V˙ pk is the direct sum of subspaces ℓ p and Pk−1 . Assertion (c) implies that for s > k the Banach spaces V pk and ℓ p are isomorphic. Since this case is not of interest, we assume in the sequel that V˙ pk satisfies the condition s(V˙ pk ) =

d ≤ k. p

(2.1)

Now let f ∈ V˙ pk and fˆ be given by fˆ(x) := lim f (y), y→x

x ∈ Qd ,

for d ≥ 2, and by ⎧ ( ) 1 ⎪ ⎪ f (x − ) + f (x + ) , 0 < x < 1, ⎪ ⎪ 2 ⎪ ⎨ fˆ(x) := f (0+ ), x = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ f (1− ), x = 1, for d = 1. Let N V˙ pk := { f ∈ V˙ pk : f = fˆ}. ˙k ˙k By the definition, N V˙ pk = ( ) C ∩ V p if d ≥ 2 and consists of functions f ∈ V p such that 1 − + f (x) = 2 f (x ) + f (x ) , x ∈ (0, 1), if d = 1. Corollary 2.2. (a) The linear map P on V˙ pk sending f to fˆ is a projection of norm 1 onto N V˙ pk . (b) The kernel of P coincides as a linear space with ℓ p and, moreover, | f |V pk ≤ ∥ f ∥ p ≤ 2| f |V pk ,

f ∈ ker(P).

In particular, (ker(P), | · |V pk ) is a Banach space. (c) V˙ pk is isomorphic to the direct sum N V˙ pk ⊕ ℓ p . Hereafter, the direct sum of (semi-) normed spaces X , Y is defined by a (semi-) norm given for (x, y) ∈ X × Y by ∥(x, y)∥ := ∥x∥ X + ∥y∥Y . Now we present two results on C ∞ approximation of V˙ pk functions. The first of them concerns approximation in the weak∗ topology of V˙ pk induced by duality ℓ∗1 = ℓ∞ . In its formulation and proof we use integrability on Q d of functions f ∈ V˙ pk , see Theorem 2.1.

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For every function f ∈ V˙ pk there is a sequence of C ∞ functions { f n }n∈N such

Theorem 2.3. that

lim ∥ f n ∥∞ ≤ 3∥ f ∥∞

n→∞

and

lim | f n |V pk ≤ 5| f |V pk ,

(2.2)

n→∞

and, moreover, for every finite signed Borel measure on Q d , ∫ lim ( f − f n ) dµ = 0. n→∞

(2.3)

Qd

As a consequence of the theorem we obtain the following generalization of the classical Helly selection theorem. Corollary 2.4. For every bounded sequence { f n }n∈N ⊂ V˙ pk there are sequences J ⊂ N and d { pn }n∈J ⊂ Pk−1 such that the sequence { f n + pn }n∈J converges pointwise to some f ∈ V˙ pk and, moreover, | f |V pk ≤ lim | f n |V pk . n∈J

The next result used in the proof of the second approximation result relates the spaces V˙ pk k and N V˙ pk to the space V˙ p∞ defined in the very same way as V˙ pk but with ℓ∞ replaced by L ∞ and with the respectively modified variation, see Definition 1.9. In its formulation, L : V˙ pk → L ∞ is a linear map sending a function from V˙ pk to its equivalence class in L ∞ . k k Theorem 2.5. (a) range(L) = V˙ p∞ and the operator L : V˙ pk → V˙ p∞ has norm 1; k (b) ker(L) = ker(P) = ℓ p ⊂ V˙ p . k (c) L maps N V˙ pk isometrically onto V˙ p∞ .

The second approximation result characterizes V˙ pk functions admitting C ∞ approximation, i.e., functions of the subspace V˙ kp . Let a function f ∈ V˙ pk and smoothness s of V˙ pk satisfy the condition

Theorem 2.6.

k > s (:= d/ p). Then f ∈

˙ kp , V

(2.4)

i.e., f is approximated by C

lim sup ⎝

functions in

V˙ pk ,

see (1.24), iff

⎞ 1p

⎛ ε→0 d(π)≤ε

∞

∑

E k ( f ; Q) p ⎠ = 0;

(2.5)

Q∈π

hereafter d(π ) := sup Q∈π |Q|. Remark 2.7. In fact, we will see that C ∞ ⊂ V˙ pk and V˙ kp ⊂ C for s ≤ k. 2.2. Relations between multivariate BV spaces and Lipschitz–Besov spaces Embeddings connecting Jordan–Wiener spaces BV p and Lipschitz spaces over L p were discovered by Hardy and Littlewood [27] and then intensively studied by many researches, see, in particular, [24,29] and references therein.

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Below we present relations of these and some other types whose proofs however require essentially new tools including Sobolev type embeddings, interpolation space results and a multivariate generalization of the Whitney type inequality. We begin with the last result where we deal with the definition of (k, p)-variation via k-oscillation, see (1.21). d For every f ∈ ℓloc ∞ (R ) the following two-sided inequality ⎫1 ⎧ ⎬p ⎨∑ p osck ( f ; Q) ≈ sup ⎭ π ∈Π ⎩

Theorem 2.8. | f |V pk

(2.6)

Q∈π

holds with constants of equivalence depending only on k, d. Hereafter Π stands for the set of all finite packings in Q d . Remark 2.9. The main point of this result is a generalized Whitney inequality given by E k ( f ; Q) ≤ c(k, d) osck ( f ; Q).

(2.7)

d The result is known for functions from L loc p (R ), 1 ≤ p ≤ ∞, and many other Banach function spaces [5] but its proof exploits tools of measure theory which is inapplicable for the derivation of (2.6).

The next result shows that a V pk function of smoothness s ∈ (0, k] “almost” coincides with a Lipschitz function of the same smoothness. Here the Lipschitz space Λk,s consists of continuous functions on Q d satisfying for x, x + kh ∈ Q d the inequality |∆kh f (x)| ≤ c∥h∥s ,

(2.8)

where ∥h∥ := max1≤i≤d |h i |, h = (h 1 , . . . , h d ) ∈ R . The equality d

| f |Λk,s := inf c

(2.9)

defines a Banach seminorm of Λk,s . Moreover, Λk,s can be introduced using continuous derivatives and differences of order at most two. In fact, setting r = r (s) := max{n ∈ Z+ ; n < s}

(2.10)

we have the following result, see, e.g., [10, Thm. 2.32]. Proposition 2.10. (a) Let s ∈ (0, k) be a noninteger or s = k, then Λk,s = C r Li p s−r . σ

(2.11)

Here Li p := Λ , 0 < σ ≤ 1. In particular, Λk,k = C k−1,1 . 1,σ

(b) Let s ∈ (0, k) be an integer, then Λk,s = C s−1 Z = C r Λ2,1 ,

(2.12)

where Z := Λ2,1 denotes Zygmund space. In both cases, | f |C r Λk,s := max |D α f |Λk,s . |α|=r

(2.13)

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The next result goes back to Luzin’s classical theorem on approximation of measurable functions on [0, 1] by continuous functions in a metric given for f, g ∈ [0, 1] → R by ρ( f, g) := |{x ∈ [0, 1] ; f (x) ̸= g(x)}|. Theorem 2.11. Let f belong to the space V˙ pk of smoothness s ∈ (0, k] and ε ∈ (0, 1). The following is true. (a) If s < k, then there is a function f ε ∈ Λk,s such that ρ( f, f ε ) < ε.

(2.14)

(b) If s = k, then there is a function f ε ∈ C k such that (2.14) is fulfilled. Using some facts established in the proof of this theorem we will then study pointwise behavior of V˙ pk functions. Model cases for this are, respectively, Lebesgue’s differentiation theorem for BV functions and the Marcinkiewicz theorem [34] asserting that f ∈ BV p satisfies the pointwise Lipschitz condition 1

lim h − p | f (x + h) − f (x)| < ∞

h→0

for almost each x ∈ [0, 1]. To formulate the corresponding multivariate results we use the following notions. Definition 2.12. if

d k,s A function f ∈ ℓloc ∞ (R ) belongs to a one-pointed Lipschitz class Λ (x 0 )

lim r −s osck ( f ; Q r (x0 )) < ∞;

(2.15)

r →0

here k ∈ N, 0 < s ≤ k and x0 ∈ Q˚ d . Similarly to the definition of functions from Λk,s we use the notations Li p s (x0 ) := Λ1,s (x0 ),

Z (x0 ) := Λ2,1 (x0 ).

Further, we define the so-called Taylor spaces introduced by Peano [39] for d = 1 and d studied (and named) by Calder´on and Zygmund [15] for functions from L loc p (R ), 1 ≤ p < ∞. d s Definition 2.13. (a) A function f ∈ ℓloc ∞ (R ) belongs to the space T (x 0 ), s > 0, if there is a (Taylor) polynomial Tx0 ( f ) ∑ f α (x0 ) Tx0 ( f ; x) := (x − x0 )α (2.16) α! |α|
such that | f − Tx0 ( f ; x)| = O(∥x − x0 ∥s )

as

x → x0 .

(2.17)

d s (b) If, moreover, s is integer, then f ∈ ℓloc ∞ (R ) belongs to the space t (x 0 ) whenever there exists a polynomial Tx0 ( f ) of degree s such that

| f − Tx0 ( f ; x)| = o(∥x − x0 ∥s )

as

x → x0 .

(2.18)

A. Brudnyi and Y. Brudnyi / Journal of Approximation Theory 251 (2020) 105346

15

Remark 2.14. (a) It is readily seen that the polynomials Tx0 ( f ) are unique. (b) If f ∈ t k (x0 ), then the kth homogeneous part of Tx0 ( f ) ∑ f α (x0 ) Dxk0 ( f ; x) := (x − x0 )α α! |α|=k is called the Peano kth differential at x0 . It is the matter of definition to check that the kth differential in the classical sense ∑ D α f (x0 ) (x − x0 )α ∂xk0 f (x) := α! |α|=k coincides with that of Peano for k = 1. However, this is, in general, incorrect for k ≥ 2. In particular, Denjoy [21] proved that for every nowhere dense closed set S ⊂ [0, 1] and k ≥ 2 there is a function f whose kth derivative in the classical sense exists only at points of the completion S c of S while the Peano kth derivative exists at all points of [0, 1]. The converse to this assertion that also proved in [21] leads in the multivariate case to the following. Conjecture. If the Peano kth differential Dxk f , k ≥ 2, exists at every point of x ∈ Q d , then kth differential ∂xk f in the classical sense exists a.e. (c) Unlike the classical case, the Peano definition can be extended to a much more complicated setting, see, in particular, [8] where this is done for multivariate functions on weak Markov sets including Ahlfors a-regular sets in Rd with d − 1 < a ≤ d and fractals of such complicated structure as the von Koch curves, the Cantor sets etc. Such kind of generalization has useful applications in harmonic analysis and singular integral operators theory, see [15,35] and subsequent papers. Theorem 2.15. Let a function f belong to the space V˙ pk of smoothness s := the following is true for almost each x0 ∈ Q d . (a)

d p

∈ (0, k]. Then

f ∈ Λk,s (x0 ); (b) If s = ℓ + λ < k, where ℓ ∈ Z+ and 0 < λ ≤ 1, then f ∈ T s (x0 ); (c) If s = k, then f ∈ t k (x0 ). Finally, we present results connecting V˙ pk spaces with Lipschitz–Besov spaces over L p , 1 ≤ p < ∞. The latter family of spaces is denoted by {Λks pq }, where k ∈ N, 0 < s ≤ k and 1 ≤ p, q < ∞. For its definition we need the following: Definition 2.16. The (k, p)-modulus of continuity is a function L p × R+ → R+ given for f ∈ L p and 0 < t ≤ 1 by ωkp ( f ; t) := sup ∥∆kh f ∥ L p (Q d ) , ∥h∥≤t

kh

(2.19)

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where ∥h∥ := max1≤i≤d |h i | and Q dkh := {x ∈ Q d : x + kh ∈ Q d }.

(2.20)

is the domain of the function x ↦→ (∆kh f )(x). Now the space Λks pq is defined for q < ∞ by the seminorm (∫ | f |Λkspq :=

1

0

(

ωkp ( f ; t) ts

)q

dt t

) q1 (2.21)

and for q = ∞ by | f |Λksp∞ := sup t>0

ωkp ( f ; t) . ts

(2.22)

For the special case q = ∞, s = k we also use the notation Li p kp := Λkk p∞ .

(2.23)

Remark 2.17. (1) We also use the introduced notions for functions on a subcube Q ⊂ Q d . In this case, f ∈ L p (Q) and the corresponding notions are denoted by ωkp ( f ; t; Q), Λks pq (Q) k and Li p p (Q). ˙s (2) The space Λks pq with k − 1 ≤ s < k coincides with the homogeneous Besov space B pq d ks while the space Λ pq /Pk−1 up to equivalence of norms coincides with the Besov space B spq , see, e.g., [45, Sec.II.2.3]. (3) In the case of one variable, the Hardy–Littlewood theorem [27] asserts (in our notations) that 1

1

Li p p ⊂ N V˙ p ↪→ Li p pp for 1 < p < ∞ and Li p 1 ⊂ N V˙ ∼ = Li p11 . In more details, let L be a linear operator sending a Lebesgue integrable function to its class of equivalence in L 1 . Then the cited result asserts that the trace L| N V˙ p is a continuous injection 1/ p into Li p p if 1 < p < ∞ and a continuous bijection onto Li p11 if p = 1. Theorem 2.18. (a) If s :=

d p

∈ (0, k], then the operator L| N V˙ pk gives the map

N V˙ pk ↪→ Λks p∞ .

(2.24) Li p kp .)

(Let us recall that if s = k the right-hand side is denoted by Moreover, if 0 < s < k, then there is a linear continuous operator right inverse to L| N V˙ pk that gives the map ˙k Λks p1 ↪→ N V p .

(2.25)

(b) If s = k = d, hence, p = 1, then L| N V˙ d gives the linear isomorphism 1

N V˙1d ∼ = Li p1d .

(2.26)

Let us note that embedding (2.24) and isomorphism (2.26) give the above mentioned Hardy–Littlewood results while embedding (2.25) is apparently new even for this case.

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Remark 2.19. The left-side Hardy–Littlewood embeddings also can be extended to the multivariate case, we have d Λk,s ⊂ V˙ pk , s := ∈ (0, k]. (2.27) p In fact, by definition, see (2.8), (2.9), s

osck ( f ; Q) ≤ |Q| d | f |Λk,s ,

Q ⊂ Qd ,

while by Theorem 2.8 | f |V pk is bounded from above by sup π

⎧ ⎨∑ ⎩

osck ( f ; Q) p

⎫1 ⎬p ⎭

Q∈π

≤ sup π

⎧ ⎨∑ ⎩

|Q|

sp d

Q∈π

⎫1 ⎬p

| f |Λk,s = | f |Λk,s .

⎭

As a consequence of Theorem 2.18 (b) we present the multivariate generalization of the classical Lebesgue–Vitali theorem on absolutely continuous functions formulated as property (d) of the space V˙ := V˙1d in Section 1.1. Let us recall that AC denotes the space of functions f ∈ V˙ satisfying the condition ∑ lim sup oscd ( f ; Q) = 0; (2.28) ε→0 |π |≤ε

here |π | :=

∑

Q∈π Q∈π

|Q| and oscd can be replaced by E d , see Theorems 2.8 and 6.1.

Corollary 2.20. The map sending a function from V˙ to its equivalence class in L 1 gives rise to the isomorphism AC ∼ = W˙ 1d .

(2.29)

Moreover, there are a continuous linear integral operator J : L 1 (Q d ; R N ) → C and a d linear projection P : L 1 → Pd−1 such that for every f ∈ AC f = P f + J (∇ d f );

(2.30)

here ∇ d f := (D α f )|α|=d is the tuple of partial derivatives of order d of f and N := card{α ∈ Zd+ : |α| = d}. 2.3. Duality theorems Now we introduce a predual to V pk space denoted by U pk and describe its properties. The space under consideration is constructed by the following building blocks called (k, p)-atoms. Definition 2.21.

A function a ∈ ℓ1 (= ℓ1 (Q d )) is said to be a (k, p)-atom if it is such that

(a) a is supported by a subcube Q ⊂ Q d ; (b) ∥a∥ ∑1 ≤ 1; (c) x α a(x) = 0 for all |α| ≤ k − 1. x∈Q d

The subject of the definition is denoted by a Q .

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Definition 2.22 ((k, p)-chain). A function b ∈ ℓ1 is said to be a (k, p)-chain subordinate to a packing π ∈ Π if b belongs to the linear hull of the family {a Q } Q∈π . The subject of the definition is denoted by bπ . Moreover, we set ⎞ 1′ ⎛ p ∑ ′ [bπ ] p′ := ∥{c Q } Q∈π ∥ p′ := ⎝ |c Q | p ⎠

(2.31)

Q∈π

whenever bπ =

∑

cQ aQ ,

{c Q } Q∈π ⊂ R.

(2.32)

Q∈π

(As usual, 1p + p1′ = 1.) Further, we define a dense subspace of the required predual space denoted by (U pk )0 by setting (U pk )0 := linhull(Akp ),

(2.33)

Akp

where is the family of all (k, p)-atoms. Equivalently, (U pk )0 := linhull(B kp ),

(2.34)

where B kp is the family of all (k, p)-chains. Moreover, we equip (U pk )0 with a seminorm5 f ↦→ ∥ f ∥U pk given for f ∈ (U pk )0 by ∑ ∥ f ∥U pk := inf [bπ ] p′ ,

(2.35)

where infimum is taken over all presentations ∑ f = bπ .

(2.36)

π

π

Definition 2.23.

The space U pk is completion of the space ((U pk )0 , ∥ · ∥U pk ).

Now we describe properties of the space U pk . k In the formulation of the result, we use a space U p∞ defined by the very same construction k as U p but with ℓ1 replaced by L 1 . Hence, in∫this case a Q is an L 1 function supported by a d subcube Q orthogonal to Pk−1 and such that Q d |a Q |, d x ≤ 1. It follows from [9, Th. 2.6] that k k (U p∞ )∗ ≡ V p∞ .

Theorem 2.24. (a) U pk is a Banach space. (b) B(U pk ) is the closure of the symmetric convex hull of the set {bπ ∈ (U pk )0 : [bπ ] p′ ≤ 1}. (c) The space U pk is nonseparable and contains a separable subspace, say, Uˆ pk , such that k Uˆ pk ≡ U p∞

and

(U pk /Uˆ pk )∗ ∼ = ℓp.

d Finally, we present the duality theorem for the Banach space V pk (:= V˙ pk /Pk−1 , see (1.22)). 5

In fact, norm, see Theorem 2.24 (a) below.

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Theorem 2.25. (U pk )∗

≡

19

It is true that

V pk .

(2.37)

In more details, every linear functional f ∈ (U pk )∗ is assigned at u ∈ (U pk )0 by ∑ ⟨ f, u⟩ := v f (x)u(x),

(2.38)

x∈Q d

where v f is a uniquely defined function of V˙ pk linearly depending on f . Moreover, it is true that ∥ f ∥(U pk )∗ = |v f |V pk .

(2.39)

2.4. Two stars theorems Now we present duality results connecting the space Vkp with the spaces V pk and U pk . In their ˜pk plays an essential role. In its definition, we proofs, an extension of the space U pk denoted by U use the Banach space M of finite signed Borel measures on Q d equipped with a norm given for µ ∈ M by ∥µ∥ M := var µ.

(2.40)

Qd

˜pk as a direct analog of the space U pk using as building blocks atoms {µ Q } Q⊂Q d ⊂ M We define U satisfying the conditions (i) supp µ ⊂ Q; (ii) ∫ ∥µ∥ M ≤ 1; x α dµ = 0 for all |α| ≤ k − 1. (iii) Qd

˜pk )0 and by U ˜pk is denoted The corresponding linear hull of the set {µ Q } Q⊂Q d is denoted by (U k 0 its completion with respect to a seminorm defined similarly to that for (U p ) . ˜pk is Repeating line by line the proof of Theorem 2.24 (a) we will show that the space U Banach. Since every chain b ∈ (U pk )0 can be identified with a discrete measure µb ∈ M with ˜pk )0 . µb ({x}) := b(x), x ∈ Q d , the space (U pk )0 is embedded (as a vector space) in (U ˜pk )0 , b ↦→ µb , extends to an isometric embedding of Theorem 2.26. (a) The map (U pk )0 → (U k k ˜p denoted by I. U p into U ˜pk onto U pk denoted by E such that (b) There exists a linear continuous surjection of U ˜pk )0 = {0} ker(E) ∩ (U

and

E ◦ I = id.

˜pk )0 ⊂ M with a dense subspace of U pk containing (U pk )0 and The result allows to identify (U to apply the Riesz representation theorem in the forthcoming proofs. Now we formulate a result giving another presentation of the predual space U pk and derive from here the mentioned two stars theorem. Theorem 2.27.

If the space V pk has smoothness s < k and p > 1, then

(Vkp )∗ ∼ = U pk .

(2.41)

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In more details, every linear continuous functional on Vkp is of the form Tu ( f ) = f (u),

f ∈ Vkp ,

where u ∈ U pk is a uniquely defined element such that ∥Tu ∥(Vkp )∗ ≈ ∥u∥U pk with constants of equivalence independent of u. As a direct consequence we have the following Corollary 2.28 (Two Stars Theorem). If V pk satisfies the conditions s < k and p > 1, then (Vkp )∗∗ ∼ = V pk .

(2.42)

3. Proofs of Theorem 2.1 and Corollary 2.2 3.1. Auxiliary results Here we prove some auxiliary results that are interesting in their own right. Lemma 3.1. Let {Q i }i∈N be the family of nonoverlapping cubes in Q d . Then for each f ∈ V˙ pk the limit lim E k ( f ; Q i ) exists and equals 0. i→∞

Proof. By the definition of the seminorm on V˙ pk , see (1.20), we have ∞ ∑

p

E k ( f ; Q i ) p ≤ | f |V k < ∞. p

i=1

Since 1 ≤ p < ∞, this implies the required statement. □ Further, let Qx denote the family of subcubes of Q d containing a point x ∈ Q d . We set for a function f ∈ V˙ pk , f k (x) :=

lim

Qx ∋Q→{x}

E k ( f ; Q);

(3.1)

here the convergence is in the Hausdorff metric on subsets of Rd . Clearly, 0 ≤ f k (x) ≤ | f |V pk , x ∈ Q d . In fact, the following result holds. Lemma 3.2.

The function f k belongs to ℓ p and satisfies the inequality ∥ f k ∥ p ≤ | f |V pk .

Proof. Assume that c is a non-isolated point of range( f k ) := f k (Q d ). Then there exists a sequence of distinct points {xi }i∈N ⊂ Q d such that limi→∞ f k (xi ) = c. In turn, by the definition of f k , there is a family of nonoverlapping cubes Q i ⊂ Qxi such that E k ( f ; Q i ) ≥ i−1 f k (xi ), i i ∈ N. Hence, due to Lemma 3.1, 0 ≤ c = lim f k (xi ) ≤ lim E k ( f ; Q i ) = 0. i→∞

i→∞

Hence, the set range( f k ) \ {0} consists of isolated points only and is at most countable. Next, if c ∈ range( f k ) \ {0}, then the set f k−1 (c) ⊂ Q d is finite. (In fact, otherwise there is a family of nonoverlapping cubes {Q i }i∈N such that limi→∞ E k ( f ; Q i ) = c ̸= 0 which contradicts Lemma 3.1).

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( ) These, in turn, imply that supp f k := f k−1 range( f k ) \ {0} is at most countable. Moreover, ∥ f k ; supp f k ∥ p ≤ | f |V pk , by definitions of f k and the seminorm on V˙ pk . □ Proposition 3.3.

Let x ∈ Q d . Then f k (x) = 0 iff f is continuous at x.

Proof. Suppose that x ∈ Q d is such that f k (x) = 0. We prove that f is continuous at x. Since f ∈ ℓ∞ , each infinite subset of range( f ) has a limit point. Hence, we have to prove that if {xi }i∈N ⊂ Q d is a sequence converging to x and the sequence { f (xi )}i∈N converges to some c ∈ R, then c = f (x). To this end, passing to a subsequence of {xi }i∈N with more rapid convergence to x, if necessary, without loss of generality we can assume that lim

i→∞

∥xi+1 − x∥ℓd∞ ∥xi+1 − xi ∥ℓd∞

= 0.

(3.2)

Since the projections of the interval [x, xi ] ⊂ Q d onto the coordinate axes have lengths bounded from above by ∥x − xi ∥ℓd∞ (≤ 1), there exists a cube Q i ∈ Qx of sidelength ∥x − xi ∥ℓd∞ containing [x, xi ]. In particular, the sequence {Q i }i∈N converges to {x} in the Hausdorff metric. Moreover, sets Q i ∪ Q i+1 are connected so that their projections on the coordinate axes are compact subintervals of [0, 1] of lengths ≤ ∥x − xi ∥ℓd∞ + ∥x − xi+1 ∥ℓd∞ . Hence, there exists a cube Q i,i+1 ∈ Qx of sidelength ≤ ∥x −xi ∥ℓd∞ +∥x −xi+1 ∥ℓd∞ containing Q i ∪ Q i+1 . In particular, Q i,i+1 → {x} in the Hausdorff metric as x → ∞. d , i ∈ N, be such that Now let m i,i+1 ∈ Pk−1 E k ( f ; Q i,i+1 ) = ∥ f − m i,i+1 ; Q i,i+1 ∥∞ . Since ∥m i,i+1 ; Q i,i+1 ∥∞ ≤ ∥ f ∥∞ and Q i,i+1 contains the interval [xi , xi+1 ], the mean-value theorem and the A. Markov inequality for derivatives of polynomials, see, e.g., [43, IV.8], give for a positive constant c := c(k, d) | f (x) − f (xi+1 )| ≤ | f (x) − m i,i+1 (x)| + | f (xi+1 ) − m i,i+1 (xi+1 )| + |m i,i+1 (x) − m i,i+1 (xi+1 )| ≤ 2E k ( f ; Q i,i+1 ) + max ∥∇m i,i+1 ∥ℓd∞ Q i,i+1

≤ 2E k ( f ; Q i,i+1 ) +

≤ 2E k ( f ; Q i,i+1 ) +

c ∥ f ∥∞ ∥x − xi+1 ∥ℓd∞ diam(Q i,i+1 ) c∥x − xi+1 ∥ℓd∞ ∥xi − xi+1 ∥ℓd∞

∥ f ∥∞ . (3.3)

Further, since f k (x) = 0, condition (3.1) implies that lim

Qx ∋Q→{x}

E k ( f ; Q) = 0.

From here and (3.2) we obtain that the right-hand side of (3.3) tends to 0 as i tends to ∞. Thus, c = limi→∞ f (xi+1 ) = f (x), as required, i.e., f is continuous at x.

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Conversely, suppose that f ∈ V˙ pk is continuous at x ∈ Q d . Then 0 ≤ f k (x) := ≤

lim

Qx ∋Q→{x}

lim

Qx ∋Q→{x}

E k ( f ; Q) ≤

lim

Qx ∋Q→{x}

E 1 ( f ; Q)

∥ f − f (x); Q∥∞ = 0.

Hence, f k (x) = 0. This completes the proof of the proposition.

□

Proposition 3.4. Let ℓ be a line intersecting Q d by a nontrivial interval I . Then for each f ∈ V˙ pk its restriction to I belongs to the space V˙ pk (I ). Proof. We need the following geometric result. Lemma 3.5. Let {I j }1≤ j≤s be a family of nonoverlapping nontrivial closed intervals in I . There exists a family of nonoverlapping cubes {Q j }1≤ j≤s in Q d such that 1

|Q j | d = ℓ∞ (I j )

and

Q j ∩ I = Ij,

1 ≤ j ≤ s;

(3.4)

here ℓ∞ ([x, y]) := max {|xi − yi |} is the ℓd∞ -diameter of [x, y] ⊂ Rd . 1≤i≤d

Proof. Let αi be the angle between a vector parallel to I and the coordinate vector ei , 1 ≤ i ≤ d. Renumerating the coordinates axes, if necessary, without loss of generality we assume that the sequence {| cos αi |}1≤i≤d is nonincreasing. Then the lengths of the orthogonal projections of every I j on the coordinate axes xi , 1 ≤ i ≤ d, also form a nonincreasing sequence. Using this we proceed by induction on d starting from the trivial case d = 1 and assuming that the result holds for all dimensions ≤ d − 1 with d ≥ 2. To prove the lemma for d ≥ 2 we use the map πd : x ↦→ (x1 , . . . , xd−1 , 0) sending Q d onto d−1 Q := {x ∈ Q d : xd = 0}. Then I ′ := πd (I ) is a closed interval in the cube Q d−1 and ′ I j := πd (I j ), 1 ≤ j ≤ s, is a family of nonoverlapping closed intervals in I ′ . Moreover, I ′ is nontrivial (hence, every I j′ is) since otherwise | cos αd | = 0 and, hence, by our assumption all | cos α j | = 0 contradicting nontriviality of I . Now by the induction hypothesis there is a family of nonoverlapping cubes Q ′j ⊂ Q d−1 , 1 ≤ j ≤ s, satisfying condition (3.4) for the family {I j′ }1≤ j≤s . Next, preimages of Q ′j with respect to the map πd | Q d form the family of nonoverlapping rectangular parallelotopes, say Q˜ j , with bases Q ′j and heights being the unit intervals parallel to the xd -axis; moreover, I j = ℓ ∩ Q˜ j , 1 ≤ j ≤ s. Let I j,d be the image of I j under the orthogonal projection onto the xd axis. Then I j,d is a (possibly trivial) interval of length ≤ ℓ∞ (I j ) ≤ 1 in the unit interval in this axes. In particular, ′ there is a subinterval I j,d in the unit interval of length ℓ∞ (I j ) containing I j,d . The intersection of the preimage of this interval under the projection onto xd with Q˜ j is clearly a cube Q j of sidelength ℓ∞ (I j ) satisfying the required statement. □ Finally, using the result and notations of Lemma 3.5, for each f ∈ V˙ pk we have E k ( f ; I j ) ≤ E k ( f ; Q j ),

1 ≤ j ≤ s.

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23

In particular, ⎞ 1p ⎛ ⎞ 1p ⎛ s s ∑ ∑ ⎝ E k ( f ; Q j ) p ⎠ ≤ | f |V pk . Ek ( f ; I j ) p ⎠ ≤ ⎝ j=1

j=1

Taking in the left-hand side supremum over all packings {I j } ⊂ I we obtain that f | I ∈ V˙ pk (I ) and | f | I |V pk (I ) ≤ | f |V pk . The proof of the proposition is complete. □ 3.2. Proof of Theorem 2.1 Proof. (a) Suppose that d = 1 and f ∈ V˙ pk ([0, 1]). We have to prove that f has one-sided limits at each point x ∈ [0, 1]. Since f ∈ ℓ∞ , each infinite subset of range( f ) has a limit point. Hence, we have to prove that if {x j,i }i∈N ⊂ [0, 1], j = 1, 2, are sequences converging to x from the same side such that sequences { f (x j,i )}i∈N converge to some c j ∈ R, j = 1, 2, then c1 = c2 . To this end, passing to subsequences of {x j,i }i∈N , j = 1, 2, with more rapid convergence to x, if necessary, without loss of generality we may assume that |x2,i+1 − x1,i+1 | = 0; (3.5) x1,i+1 < x2,i+1 < x1,i < x2,i for all i ∈ N and lim i→∞ |x 2,i+1 − x 1,i | here all intervals Ii := [x1,i , x2,i ] ⊂ [0, 1], i ∈ N, belong to the same connected component of the set R \ {x}. By definition, {Ii }i∈N ⊂ [0, 1] is a family of nonoverlapping intervals converging to {x} in the Hausdorff metric. The second condition (3.5) implies that there exists a family of nonoverlapping intervals {Ii′ }i∈N ⊂ [0, 1] such that Ii ⊂ Ii′

for all i ∈ N,

lim

i→∞

|Ii | = 0, |Ii′ |

and

lim Ii′ = {x}

i→∞

(3.6)

in the Hausdorff metric. In particular, due to Lemma 3.1, lim E k ( f ; Ii′ ) = 0.

(3.7)

i→∞

1 Let m i ∈ Pk−1 , i ∈ N, be such that

E k ( f ; Ii′ ) = ∥ f − m i ; Ii′ ∥∞ . Since ∥m i ; Ii′ ∥∞ ≤ ∥ f ∥∞ and Ii′ ⊃ Ii , the mean-value theorem and the A. Markov inequality for derivatives of polynomials give | f (x2,i ) − f (x1,i )| ≤ | f (x2,i ) − m i (x2,i )| + | f (x1,i ) − m i (x1,i )| + |m i (x2,i ) − m i (x1,i )| ≤ 2E k ( f ; Ii′ ) +

2(k − 1)2 ∥ f ∥∞ |Ii |. |Ii′ |

Since the right-hand side of the inequality tends to 0 as i → ∞, see (3.6), (3.7), c1 = lim f (x1,i ) = lim f (x2,i ) = c2 . i→∞

i→∞

This completes the proof of part (a) of the theorem.

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(b) Now suppose that f ∈ V˙ pk (Q d ) with d ≥ 2. We have to prove that f has limits at each point x ∈ Q d . First, due to Proposition 3.4, for each line ℓ through x such that I := ℓ ∩ Q d ̸= {x} the restriction f | I belongs to V˙ pk (I ). In particular, according to (a) f | I has one sided limits at x. This implies that f has radial limits at x. Next, we show that all these limits coincide. Let Hx be a hyperplane through x parallel to a coordinate hyperplane and x j be the coordinate axis orthogonal to Hx . Then Rd \ Hx is the union of two nonintersecting open half spaces S1 , S2 such that at least one of them, say S1 , intersects Q d . Lemma 3.6. lim

Let y1 , y2 ∈ S1 ∩ Q d be distinct. Then

(x,y1 ]∋z→x

f (z) =

lim

(x,y2 ]∋z→x

f (z).

(3.8)

Proof. By our assumption, the orthogonal projection onto the x j axis p j maps lines (x, yi ), i = 1, 2, bijectively onto x j . Let ( ) ( ) I := p j (x, y1 ] ∩ p j (x, y2 ] be the half open interval with the endpoint p j (x). We choose a monotonic (with respect to the natural order on x j ) sequence {z n }n∈N ⊂ I converging to p j (x) and set yn,i := p −1 j (z n ) ∩ (x, yi ),

i = 1, 2.

Passing to a subsequence of {z n }n∈N with more rapid convergence to p j (x), if necessary, without loss of generality we assume that lim

n→∞

maxi=1,2 ∥yn+1,i − x∥ℓd∞ ∥z n+1 − z n ∥ℓd∞

= 0.

(3.9)

d Next, let Pn := p −1 j ([z n , z n+1 ]) ∩ Q be the rectangular parallelotope between hyperplanes −1 d and p j (z n ) inside Q . Then Pn contains points yn+1,i , i = 1, 2, and the smallest length of projections of Pn onto coordinate axes is ∥z n+1 − z n ∥ℓd∞ . Moreover, due to (3.9),

p −1 j (z n+1 )

∥yn+1,2 − yn+1,1 ∥ℓd∞ ≤ 2 max ∥yn+1,i − x∥ℓd∞ = λn · ∥z n+1 − z n ∥ℓd∞ , i

where the sequence {λn }n∈N converges to 0. These imply that for each n such that λn < 1 there exists a cube Q n ⊂ Pn of sidelength ∥z n+1 − z n ∥ℓd∞ containing points yn+1,i , i = 1, 2. By the definition, {Q n }n∈N is a sequence of nonoverlapping cubes converging to {x} in the Hausdorff metric. Hence, according to Lemma 3.1 lim E k ( f ; Q n ) = 0.

n→∞

(3.10)

Then as in (3.3) using the mean-value theorem and the Markov inequality for derivatives of polynomials on Q n , and after that using (3.9), (3.10), we obtain for some c = c(k, d) ( ) c∥yn+1,2 − yn+1,1 ∥ℓd∞ lim | f (yn+1,1 ) − f (yn+1,2 )| ≤ lim 2E( f ; Q n ) + ∥ f ∥∞ = 0. n→∞ n→∞ ∥z n+1 − z n ∥ℓd∞ This implies (3.8). □

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25

Using Lemma 3.6 we obtain the following: Lemma 3.7.

If d ≥ 2, then all radial limits of f ∈ V˙ pk (Q d ) at a point x ∈ Q d coincide.

Proof. First, observe that Q d \ {x} is covered by the union of open sets Rd \ Hxi , 1 ≤ i ≤ d, where Hxi is a hyperplane through x orthogonal to the xi axes; this is true because ⋂ Hxi = {x}. 1≤i≤d

Let {S j }1≤ j≤k , d ≤ k ≤ 2d, be the family of all connected components (open half spaces) of the sets Rd \ Hxi , 1 ≤ i ≤ d, intersecting Q d . According to Lemma 3.6, f has the same radial limit at x along each ray in S j emanating from x denoted by L j . Clearly, ⋃ Q d \ {x} ⊂ Sj. 1≤ j≤k

In particular, for a path γ : [0, 1] → Q d \ {x} there is a subset J ⊂ {1, . . . , k} such that ⋃ γ ([0, 1]) ⊂ Sj and S j ∩ γ ([0, 1]) ̸= ∅, j ∈ J. j∈J

Thus, the function F f (t) :=

lim

(x,γ (t)]∋z→x

f (z),

t ∈ [0, 1],

attains value L j on the open set γ −1 (S j ), j ∈ J . Since these open sets cover the connected interval [0, 1], the function F f must be constant. Hence, f has the same radial limits at x along the rays emanating from x passing through γ (0) and γ (1). Finally, since Q d \ {x} is path connected, every two its points are joined by a path. Together with the previous argument this completes the proof of the lemma. □ To proceed with the proof of the theorem, we require the following geometric result. Proposition 3.8. Let {Rn }n∈N ⊂ Rd be a sequence of rays with the origin at x ∈ Q d converging to a ray R. If Rn ∩ Q d \ {x} ̸= ∅ for all n, then R ∩ Q d \ {x} ̸= ∅ as well. Here convergence is defined by the points of intersection of the rays with the unit sphere centered at x. Proof. We apply induction on dimension. For d = 1 the result trivially holds. Assuming that it is valid for all dimensions ≤ d − 1, d ≥ 2, we prove it for d. To this end, consider intervals In := [x, xn ] := Rn ∩ Q d , n ∈ N. By the assumption of the lemma ∥xn − x∥ℓd∞ > 0. Passing to a subsequence of {Rn }n∈N , if necessary, we assume without loss of generality that all xn are contained in the same face, say F, of Q d and the sequence {xn }n∈N converges to a point x˜ ∈ F.

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First, let us consider x ̸∈ F. Then 0 < δ =: dist∞ (x, F) ≤ lim ∥x − xn ∥ℓd∞ = ∥x − x∥ ˜ ℓd∞ . n→∞

In particular, (x, x] ˜ is a nontrivial interval in R, i.e., R ∩ Q d ̸= ∅ in this case. Now let x ∈ F. Then all intervals In are contained in F as well. This reduces the problem to the case of the (d − 1)-dimensional cube F and the sequence of rays {Rn }n∈N such that Rn ∩ F \ {x} ̸= ∅ for all n. Clearly, the limit ray R belongs to the hyperplane containing F and, hence, by the induction hypothesis R ∩ F \ {x} ̸= ∅. This proves the proposition. □ Finally, let us complete the proof of Theorem 2.1 (b). To this end, it remains to prove for f ∈ V˙ pk that if {xn } ⊂ Q d \ {x} is a sequence converging to x such that limn→∞ f (xn ) exists, then the limit coincides with the radial limit of f at x, see Lemma 3.7. Passing to a subsequence of {xn }n∈N , if necessary, we assume without loss of generality that the rays Rn with the origin at x passing through xn , n ∈ N, converge to a ray R. By Proposition 3.8 R ∩(Q d \{x}) ̸= ∅. Then there is a hyperplane Hx passing through x orthogonal to a coordinate axis and an open half space S ⊂ Rd with the boundary Hx such that R\{x} ⊂ S. We show that for all sufficiently large n points xn belong to S as well. In fact, otherwise the corresponding rays Rn are located in the closed half space Rd \ S with the boundary Hx , hence, R as the limit of the sequence {Rn }n∈N is contained in Rd \ S as well, a contradiction. Now, we proceed as in the proof of Lemma 3.6. Specifically, we draw hyperplanes through points xn parallel to Hx . They intersect R at some points yn . Then passing to a subsequence {x j } j∈J ⊂N of {xn }n∈N we construct a sequence of nonoverlapping cubes {Q j } j∈J ⊂ S ∩ Q d \{x} converging to {x} in the Hausdorff metric such that [x j , y j ] ⊂ Q j , j ∈ J , and ∥x j − y j ∥ℓd∞ lim = 0. J ∋ j→∞ diam(Q j ) Using these and Lemma 3.1 and arguing as in the proof of Lemma 3.6 (i.e., applying the mean-value theorem and Markov inequality) we obtain that lim | f (x j ) − f (y j )| = 0.

J ∋ j→∞

This shows that limn→∞ f (xn ) coincides with the radial limit of f at x and completes the proof of part (b) of the theorem. (c) We have to prove that if s > k, then the vector space V˙ pk is the direct sum of subspaces d ℓ p and Pk−1 . Due to parts (a) and (b) of the theorem, each f ∈ V˙ pk is bounded and Lebesgue measurable. In particular, the map L : V˙ pk → L ∞ which sends a function to its equivalence class in L ∞ is well defined. Lemma 3.9.

k k It is true that range(L) ⊂ V˙ p∞ and the norm of L : V˙ pk → V˙ p∞ is ≤ 1.

k Proof. Let us recall that V˙ p∞ is the subspace of L ∞ functions on Q d equipped with the seminorm ⎛ ⎞ 1p ∑ | f |V p∞ := sup ⎝ E k∞ ( f ; Q) p ⎠ , (3.11) k π ∈Π

Q∈π

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27

where E k∞ ( f ; Q) :=

inf ∥ f − m∥ L ∞ (Q) .

d m∈Pk−1

Since for each Lebesgue measurable function f ∈ ℓ∞ , E k∞ ( f ; Q) ≤ E k ( f ; Q), the required k statement follows directly from definitions of seminorms of V˙ pk and V˙ p∞ . □ Further, since s > k, Lemma 3.1 in [9] implies that each L( f ) equals a polynomial, say d d m f ∈ Pk−1 , a.e. on Q d . Since by the definition of the map L for every m ∈ Pk−1 function d d k k k L(m) equals m a.e. on Q , map L : V˙ p → V˙ p∞ is surjective and L|P d : Pk−1 → V˙ p∞ is k−1 d k an isomorphism; hence, ker(L) ∩ Pk−1 = {0}. This implies that V˙ p is isomorphic to the direct d sum of ker(L) and Pk−1 . To complete the proof of part (c) it remains to show the following. Lemma 3.10.

ker(L) coincides with ℓ p .

Proof. First, let us show that ℓ p ⊂ V˙ pk . In fact, we have for f ∈ ℓ p ⎞ 1p

⎛ | f |V pk := sup ⎝ π∈Π

∑

⎛

E k ( f ; Q) p ⎠ ≤ ⎝

Q∈π

} p ⎞ 1p

{ ∑ Q∈π

⎠ ≤ ∥ f ∥p,

sup | f | Q

i.e., f ∈ V˙ pk as required. Clearly, L( f ) = 0 for each f ∈ ℓ p , i.e., ℓ p ⊂ ker(L). Conversely, if f ∈ ker(L), then f equals 0 a.e. on Q d and, moreover, this and Lemma 3.1, 3.2 imply that supp f is an at most countable subset of Q d . Let S ⊂ supp f be a finite subset. By definition the function { 0 if y ∈ S f S (y) := f (y) if y ̸∈ S is continuous at each point of S. Due to Proposition 3.3 we then have, see (3.1), ( f − f S )k (y) ≤ f k (y) + ( f S )k (y) = f k (y)

for all

(3.12)

y ∈ S.

On the other hand, f − f S is supported by S and ( f − f S )(y) = f (y) for each y ∈ S. The direct computation of the limit in (3.1) gives for this case ( f − f S )k (y) := lim E k ( f − f S ; Q) = Q→{y}

1 | f (y)|, 2

(3.13)

y ∈ S.

Now (3.12), (3.13) and Lemma 3.2 imply that ⎛ ⎞ 1p 1 ⎝∑ | f (y)| p ⎠ = ∥( f − f S )k ∥ p ≤ ∥ f k ∥ p ≤ | f |V pk . 2 y∈S This shows that f ∈ ℓ p and ∥ f ∥ p ≤ 2| f |V pk and implies that ker(L) ⊂ ℓ p . The proof of Theorem 2.1 is complete. □

□

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3.3. Proof of Corollary 2.2 Proof. (a) We begin with the case of d ≥ 2. Then by definition fˆ(x) := lim f (y), x ∈ Q d , y→x

and P maps f ∈ V˙ pk to fˆ. We have to prove that P is a linear projection of norm 1 onto the closed subspace C ∩ V˙ pk denoted by N V˙ pk . First we show that for f ∈ V˙ pk the function fˆ ∈ V˙ pk and | fˆ|V pk ≤ | f |V pk . Since clearly P fˆ = fˆ and C ∞ ⊂ N V˙ pk (see the proof of Theorem 2.6 in Section 5), this will prove the corollary for this case. To this end, it suffices to prove that for each closed cube Q ⊂ Q d E k ( fˆ; Q) ≤ E k ( f ; Q). (3.14) d Let m ∈ Pk−1 be such that

E k ( f ; Q) = ∥ f − m; Q∥∞ .

(3.15)

Since by Theorem 2.1 function fˆ is continuous on Q d and f − fˆ has at most countable support, there are x ∈ Q d such that | fˆ(x) − m(x)| = ∥ fˆ − m; Q∥∞ (3.16) and a sequence {xn }n∈N ̸⊂ supp( f − fˆ) in Q converging to x. Hence, due to continuity of fˆ, lim | fˆ(xn ) − m(xn )| = | fˆ(x) − m(x)|.

(3.17)

n→∞

Using (3.15)–(3.17), and the definition of fˆ we obtain E k ( fˆ; Q) ≤ ∥ fˆ − m; Q∥∞ = lim | fˆ(xn ) − m(xn )| = lim | f (xn ) − m(xn )| ≤ E k ( f ; Q) n→∞

n→∞

proving (3.14) and assertion (a) in this case. d Now let d = 1, hence, m in (3.15) belongs to Pk−1 and Q is a closed interval in [0, 1]. ( ) ˆ Moreover, f (x) := f (x) if f is a point of continuity for f , and fˆ(x) := 21 f (x − ) + f (x + ) for the remaining at most countable set of points x in (0, 1) (with the corresponding correction for x = 0, 1). As in the previous case, we show that (3.14) holds for d = 1. Let {xn }n∈N ⊂ Q be such that lim | fˆ(xn ) − m(xn )| = ∥ fˆ − m; Q∥∞ . (3.18) n→∞

If {xn }n∈N contains an infinite subsequence of points of continuity of fˆ, then we can slightly change the latter to get a sequence outside of supp( f − fˆ) (which is at most countable) satisfying (3.18). In this case the argument of the proof of the previous case gives the required inequality (3.14). Otherwise, {xn }n∈N contains an infinite subsequence, say {xn k }k∈N , of points of discontinuity for f in (0, 1). Then by the definition of fˆ we derive from (3.18): E k ( fˆ; Q) ≤ ∥ fˆ − m; Q∥∞ = lim | fˆ(xn k ) − m(xn k )| k→∞ ⏐ ⏐ ⏐ f (xn k− ) − m(xn k ) f (xn k+ ) − m(xn k ) ⏐⏐ ⏐ = lim ⏐ + ⏐ k→∞ 2 2 ≤ ∥ f − m; Q∥∞ = E( f ; Q), as required.

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Now, let xn = 0 for all n ∈ N. Then we have by (3.18) and the definition of fˆ E k ( fˆ; Q) ≤ ∥ fˆ − m; Q∥∞ = | fˆ(0) − m(0)| := lim | f (x) − m(x)| x→0+

≤ ∥ f − m; Q∥∞ = E( f ; Q). The case of xn = 1 for all n ∈ N is considered similarly. The proof of (3.14) is complete, i.e., P : V˙ pk → N V˙ pk (⊂ V˙ pk ) is a linear projection of norm 1 also for d = 1. (b) By the definition of the map P its kernel contains functions in V˙ pk with at most countable supports. Then as in the proof of part (c) of Theorem 2.1 we obtain that ker(P) = ℓ p . Moreover, we have proved in Lemma 3.10 that for each f ∈ ℓ p | f |V pk ≤ ∥ f ∥ p ≤ 2| f |V pk .

(3.19)

This shows that the identity map embeds ℓ p into V˙ pk as a Banach subspace. (c) The statement V˙ pk ∼ = N V˙ pk ⊕ ℓ p follows directly from parts (a) and (b). The proof of the corollary is complete. □ 4. Proofs of Theorems 2.3, 2.5 and Corollary 2.4 In the proofs, we use the duality between V pk and U pk established in Theorem 2.25 and k k between V p∞ and U p∞ established in [9, Th. 2.6]. 4.1. Auxiliary results 4.1.1. d In what follows, we fix an interpolating subset S ⊂ Q d for polynomials of Pk−1 . Hence, d card S = dim Pk−1 + 1 and for each function f ∈ ℓ∞ (S) there is a unique polynomial d m f ∈ Pk−1 such that m f | S = f . The norm of the linear operator f ↦→ m f is clearly bounded by a constant c = c(S). k k Let V˙ p;S ⊂ V˙ pk denote the subspace of functions vanishing on S. Clearly, codim V˙ p;S = card S and for each f ∈ V˙ pk we have the unique representation f = m f + ( f − m f ) where k k f − m f ∈ V˙ p;S . This implies that V˙ p;S equipped with the seminorm | · |V pk is a Banach space such that k V˙ p;S ≡ V pk ,

(4.1)

d where the isometry is given by the restriction of the quotient map q : V˙ pk → V pk (:= V˙ pk /Pk−1 ) k k ∗ k ˙ ˙ to V p;S . We equip V p;S with the weak topology pulled back from that of V p by this isometry. The weak∗ topology on V pk is defined by the duality relation

V pk ≡ (U pk )∗

(4.2)

that will be proved in Theorem 2.25. k Lemma 4.1. The closed unit ball B(V˙ p;S ) is compact in the topology of pointwise convergence d on Q .

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d k Proof. Since B(V˙ p;S ) is contained in B(ℓ∞ ) = [−1, 1] Q which is compact in the topology k of pointwise convergence by the Tychonoff theorem, B(V˙ p;S ) is relatively compact in this k topology. If now { f α }α∈Λ is a net in B(V˙ p;S ) converging in the topology of pointwise k convergence to a function f ∈ ℓ∞ , then we must show that f ∈ B(V˙ p;S ). k ∗ ˙ Since by the Banach–Alaoglu theorem B(V p;S ) is weak compact, see (4.2), we may assume, passing if necessary to a subnet of { f α }α∈Λ , that { f α }α∈Λ converges in the weak∗ topology to k some f˜ ∈ B(V˙ p;S ). Let us show that f = f˜. k ). By the In fact, let δx be the evaluation functional at x ∈ Q d (i.e., δx (g) := g(x), g ∈ V˙ p;S definition of S, there exists an element cx ∈ ℓ1 (S) such that ∑ d m(x) = cx (s)m(s) for all m ∈ Pk−1 .

s∈S

δx′

∑ Hence, := δx − s∈S cx (s)δ(s) regarded as a function of ℓ1 supported by S∪{x} is orthogonal d to Pk−1 ; hence, δx′ belongs to (U pk )0 ⊂ U pk , see (2.33). k Now the weak∗ convergence of { f α }α∈Λ in B(V˙ p;S ) and its pointwise convergence to f d imply for each x ∈ Q , f˜(x) = ⟨ f, δx′ ⟩ = lim⟨ f α , δx′ ⟩ = lim f α (x) = f (x). α

α

Hence, f˜ = f , as required. □ 4.1.2. In the next proofs, we use the approximation theorem from [9, Th. 2.12] asserting that for k every g ∈ V˙ p∞ there is a sequence {gn }n∈N ⊂ C ∞ such that (1) lim |gn |V p∞ = |g|V p∞ k k ;

n→∞

(2) ∫ lim

n→∞

(g − gn )h d x = 0 Qd

for each

h ∈ L 1.

The sequence {gn }n∈N is defined as follows. Let ϕ ∈ C ∞ (Rd ) be a function supported by the unit cube [−1, 1]d such that ∫ 0 ≤ ϕ ≤ 1 and ϕ d x = 1. Rd

We denote by g 0 the extension of g ∈ L ∞ by 0 outside Q d and then define gn0 , n ∈ N, by ∫ ( y ) ϕ(y) dy, x ∈ Rd . (4.3) gn0 (x) := g 0 x − n+1 ∥y∥ d ≤1 ℓ ∞

Next, let an : Rd → Rd be the λn -dilation with the center c, where λn := center of Q d . Then an (Q d ) is a subcube of Q d centered at c Q d and distℓd∞ (∂ Q d , an (Q d )) =

1 . n+1

n−1 n+1

and c is the

(4.4)

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31

We define gn := Tn (g) ∈ C ∞ by setting gn = gn0 ◦ an .

(4.5)

4.2. Proof of Theorem 2.3 Let f ∈ V˙ pk . We have to find a sequence { f n }n∈N ⊂ C ∞ satisfying the limit relations (2.2), (2.3). Let L : V˙ pk → L ∞ be the map sending a function of V˙ pk to its equivalence class in L ∞ . Since ∥L( f )∥ L ∞ (Q) ≤ ∥ f ∥ℓ∞ (Q)

for all

f ∈ V˙ pk ,

Q ⊂ Qd ,

k k the range of L is a subspace of V˙ p∞ and the linear map L : V˙ pk → V˙ p∞ is bounded of norm ≤ 1, cf. Lemma 3.9. k Let f ∈ V˙ pk and g := L( f ) ∈ V˙ p∞ . Then there is a sequence {gn }n∈N ⊂ C ∞ satisfying conditions (1), (2) of Section 4.1.2. k Further, L maps C ∞ ⊂ V˙ pk isometrically onto C ∞ ⊂ V˙ p∞ (because E k∞ ( f ; Q) = E k ( f ; Q) for all f ∈ C ∞ , cf. the argument of the proof of Lemma 3.9). In particular, there exists a sequence { f˜n }n∈N ⊂ C ∞ ⊂ V˙ pk such that L( f˜n ) = gn for all n and

lim | f˜n |V pk = lim |gn |V p∞ = |g|V p∞ ≤ | f |V pk . k k

n→∞

Lemma 4.2.

(4.6)

n→∞

For each point of continuity x ∈ Q d of f

lim f˜n (x) = f (x).

(4.7)

n→∞

Proof. We have by definitions of L , an , gn , f˜n , see (4.3)–(4.5), ⏐ ⏐ ⏐ ⏐∫ ⏐ ⏐ ) ) ( 0( y − f (x) ϕ(y) dy ⏐⏐ f an (x) − n+1 | f˜n (x) − f (x)| = ⏐⏐ ⏐ ⏐ ∥y∥ℓd∞ ≤1 ⏐ ⏐∫ ⏐ ( ( = ⏐⏐ f an (x) − ⏐ ∥y∥ℓd∞ ≤1

y n+1

)

⏐ ⏐ ⏐ ⏐ ( − f (x) ϕ(y) dy ⏐⏐ ≤ sup ⏐ f an (x) − ⏐ ∥y∥ℓd∞ )

y ) n+1

⏐ − f (x)⏐ . (4.8)

Since an (x) → x as n → ∞, the last term in (4.8) tends to 0 as n → ∞ by continuity of f at x. The lemma is proved. □ Further, by the definition of the sequence {gn }n∈N , see (4.3), (4.5), ∥ f˜n ∥∞ ≤ ∥g∥ L ∞ (≤ ∥ f ∥∞ )

for all

n ∈ N.

(4.9)

Since by Theorem 2.1 the set S f of discontinuities of f is at most countable, the Cantor diagonal procedure gives a subsequence { f˜n k }k∈N of { f˜n }n∈N such that the limit lim f˜n k (x) ≤ ∥g∥ L ∞

k→∞

exists for all x ∈ S f .

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We define f˜ ∈ ℓ∞ as the pointwise limit of the sequence { f˜n k }k∈N (in particular, f˜ coincides with f at the points of continuity of f ). Lemma 4.3.

f˜ ∈ V˙ pk and | f˜|V pk ≤ | f |V pk .

Proof. We set for brevity h k := f˜n k ,

(4.10)

k ∈ N,

d and use the interpolating set S ⊂ Q d for Pk−1 of Section 4.1.1 to decompose h k as follows

h k = m h k + (h k − m h k ), d where m h k ⊂ Pk−1 interpolates h k on S. k The second term belongs to the space V˙ p;S consisting of all functions from V˙ pk vanishing on S so that |h k − m h k |V pk = |h k |V pk . Further, due to (4.9)

∥m h k ; S∥∞ ≤ ∥g∥ L ∞ . Hence, the sequence {m h k }k∈N is uniformly bounded on Q d and, in particular, it contains a uniformly converging subsequence. Passing to such a subsequence, if necessary, without loss of generality we will assume that {m h k }k∈N itself converges uniformly to a polynomial d . Therefore the sequence {h k −m h k }k∈N is uniformly bounded and converges pointwise m ∈ Pk−1 k ˜ to f − m ∈ ℓ∞ . According to Lemma 4.1, the closed ball B(V˙ p;S ) is compact in the topology k d ˜ of pointwise convergence on Q . Hence, f − m ∈ V p;S and by (4.6) | f˜|V pk = | f˜ − m|V pk ≤ lim |h k − m h k |V pk = lim |h k |V pk ≤ | f |V pk . k→∞

k→∞

The proof of the lemma is complete. □ Now set l := f − f˜.

(4.11)

Since f˜ coincides with f on the continuity set of f , see Lemma 4.2, function l ∈ ℓ p and by Lemma 4.3, |l|V pk ≤ 2| f |V pk . Lemma 4.4. that

Let l ∈ ℓ p . There is a sequence {ln }n∈N ⊂ C ∞ converging pointwise to l such

sup ∥ln ∥∞ ≤ ∥l∥∞ n∈N

(4.12)

and

lim |ln |V pk ≤ ∥l∥ p .

n→∞

(4.13)

Proof. Since the set of functions with finite supports is dense in ℓ p , without loss of generality we may assume that l is supported by a finite set, say {x1 , . . . , xm } ⊂ Q d . In this case, Euclidean balls of radius r centered at points xi are mutually disjoint for some r > 0. Let s jn be a C ∞ function supported by the Euclidean ball B nr (x j ) such that s ji (x j ) = 1 and 0 ≤ s ji ≤ 1, n ∈ N, 1 ≤ j ≤ m. We set m ∑ ln := l(x j )s jn . (4.14) l=1

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33

By the definition, for a nontrivial closed cube Q ⊂ Q d ⎛ ⎞ 1p ∑ E k (ln ; Q) ≤ ∥ln ; Q∥∞ ≤ ∥l; Q∥∞ ≤ ⎝ |l(xi )| p ⎠ . xi ∈Q

This implies that for all n ∥ln ∥∞ ≤ ∥l∥∞

and

|ln |V pk ≤ ∥l∥ p .

and proves the required inequalities. It remains to prove that lim (l(y) − ln (y)) = 0

for all

n→∞

y ∈ Qd .

Indeed, if y coincides with some x j , 1 ≤ j ≤ m, then ln (y) = l(y)

for all

n

and the result follows. Otherwise, l(y) = 0 and there is some n 0 ∈ N such that y ̸∈ supp ln for all n ≥ n 0 . Hence, ln (y) = 0 for such n and lim (l(y) − ln (y)) = 0

n→∞

as well. The proof is complete. □ Let {lk }k∈N ⊂ C ∞ and {h k }k∈N ⊂ C ∞ be as above, see (4.13), (4.10). Then we set f n := h n + ln ,

(4.15)

n ∈ N.

and show that { f k }k∈N is the required sequence. In fact, by (4.9), (4.11) and (4.13) lim ∥ f n ∥∞ ≤ lim ∥h n ∥∞ + lim ∥ln ∥∞ ≤ ∥ f ∥∞ + ∥ f − f˜∥∞ ≤ 3∥ f ∥∞ .

n→∞

n→∞

n→∞

This gives the first inequality (2.2). Further, by (4.6) and (4.13) lim ∥h n ∥V pk + lim ∥ln ∥V pk ≤ ∥ f ∥V pk + ∥l∥ p .

n→∞

n→∞

Moreover, applying the inequality proved in Lemma 3.10 and (4.11) we get ∥l∥ p ≤ 2|l|V pk ≤ 4| f |V pk . Combining these inequalities we finally have lim ∥ f n ∥V pk ≤ 5∥ f ∥V pk .

n→∞

This proves the second inequality (2.2). Further, by definition the sequence { f k }k∈N ⊂ C ∞ belongs to the first Baire class, converges pointwise to f and is uniformly bounded. Hence, the Rosenthal Main Theorem [41] implies that ∫ lim ( f − f k ) dµ = 0 (4.16) k→∞

Qd

for all measures µ ∈ M. This proves (2.3) and completes the proof of the theorem.

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4.3. Proof of Corollary 2.4 We retain notation of Section 4.1.1. Let { f n }n∈N ⊂ V˙ pk be bounded. We choose a subsequence { f ni }i∈N such that lim | f ni |V pk = lim | f n |V pk .

i→∞

(4.17)

n→∞

Next, we write f ni = m fni + ( f ni − m fni ), d k where m fni ∈ Pk−1 , f ni − m fni ∈ V˙ p;S , i ∈ N. By the definition,

sup | f ni − m fni |V k = sup | f ni |V pk ≤ sup | f n |V pk =: r < ∞. i∈N

p

i∈N

n∈N

k Hence, the sequence { f ni − m fni }i∈N belongs to the closed ball of V˙ p;S of radius r centered at 0 and so by Lemma 4.1 is relatively compact in the ball equipped with the topology of pointwise convergence on Q d . In addition, by Theorem 2.3 this sequence is in the first Baire class. Therefore the Rosenthal Main Theorem [41] imply that it contains a pointwise convergent subsequence { f n j − m fn j } j∈J , J ⊂ N. Due to (4.17) its limit, say f ∈ V˙ S;k p , belongs to the ball of radius limn∈N | f n |V pk . This completes the proof of the corollary.

4.4. Proof of Theorem 2.5 (a) As above, let L : V˙ pk → L ∞ be the map sending a function in V˙ pk to its equivalence k class in L ∞ . It was shown in Section 4.2 that the range of L is a subset of V˙ p∞ and the linear k k map L : V˙ pk → V˙ p∞ has norm ≤ 1. We have to check that the range of L is V˙ p∞ and that ∥L∥ = 1. k Let f ∈ V˙ p∞ . Then there is a sequence {gn }n∈N ⊂ C ∞ ⊂ V˙ pk such that sup ∥gn ∥∞ ≤ ∥ f ∥ L ∞

and

n∈N

lim |gn |V pk = | f |V p∞ k ,

n→∞

(4.18)

k and, moreover, {L(gn )}n∈N ⊂ C ∞ ⊂ V p∞ weak∗ converges to f in the weak∗ topology of L ∞ ∗ defined by the duality L 1 = L ∞ , see formulas (3.2), (3.4) and Theorem 2.13 in [9]. d Using the interpolating set S ⊂ Q d for Pk−1 of Section 4.1.1 we decompose gn in a sum

gn = m gn + (gn − m gn ), d k where m gn ⊂ Pk−1 interpolates gn on S and the second term belongs to the space V˙ p;S of k ˙ functions from V p vanishing on S so that |gn − m gn |V pk = |gn |V pk . According to (4.18) the sequence {m gn }n∈N is uniformly bounded on the interpolating set S, hence, on Q d . Passing to a subsequence of {gn }n∈N , if necessary, without loss of generality we d may assume that {m gn }n∈N converges uniformly on Q d to some m ∈ Pk−1 . Moreover, (4.18) implies that

lim |gn − m gn |V pk = | f |V p∞ k ,

n→∞

k k i.e., the sequence {gn −m gn }n∈N is bounded in V˙ p;S . Using the weak∗ topology on V˙ p;S induced k k k by the duality between V˙ p;S (≡ V p ) and U p , we find a subnet {n α }α∈Λ of N such that the subnet

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35

k {gn α − m h nα }α∈Λ weak∗ converges to some h ∈ V˙ p;S . Since, in addition, the subnet {m gnα }α∈Λ d converges uniformly on Q to m, the subnet {gn α }α∈Λ of {gn }n∈N converges pointwise to g := h + m, cf. the argument of Lemma 4.1. Moreover, this subnet is uniformly bounded in ℓ∞ by (4.18). Hence, by the Rosenthal Main Theorem [41] ∫ ∫ lim gn α dµ = g dµ α

Qd

Qd

for all measures µ ∈ M. Taking here dµ = h d x, h ∈ L 1 , we obtain that ∫ ∫ lim gn α h d x = gh d x for all h ∈ L 1 . α

Qd

Qd

The latter implies that the net {L(gn α )}α∈Λ weak∗ converges to L(g). Hence, by (4.18), L(g) = f. k This proves surjectivity of the map L : V˙ pk → V˙ p∞ . Finally, by (4.18) and the inequality ∥L∥ ≤ 1 we get |g|V pk = |L(g)|V p∞ ≤ |g|V pk , k i.e., |L(g)|V p∞ = |g|V pk . This implies that ∥L∥ = 1. k The proof of part (a) of the theorem is complete. (b) Now we have to prove that ker(L) = ker(P) = ℓ p . By the definition of the map L its kernel consists of all functions in V˙ pk which are zeros outside sets of Lebesgue measure zero. Then by Lemma 3.10 ker(L) = ℓ p . The same is established in Corollary 2.2 (b) for ker(P). This completes the proof of this part of the theorem. k (c) It remains to show that L maps N V˙ pk isometrically onto V˙ p∞ . k k ˙ ˙ As in part (a) for each f ∈ V p∞ we take g ∈ V p such that L(g) = f and |g|V pk = | f |V p∞ k . Then we set h := P(g) (∈ N V˙ pk ). Since h − g ∈ ℓ p , see Corollary 2.2, ∥L∥ = 1 and ∥P∥ = 1, L(h) = L(g) = f

and

| f |V p∞ ≤ |h|V pk ≤ |g|V pk = | f |V p∞ k . k

Moreover, ker(L) ∩ N V˙ pk = ℓ p ∩ N V˙ pk = {0} by Corollary 2.2. These imply existence for each k a unique h ∈ N V˙ pk such that L(h) = f and |h|V pk = | f |V p∞ f ∈ V˙ p∞ k . k k ˙ Thus, L : N V p → V p∞ is an isometric isomorphism of semi-Banach spaces. The proof of Theorem 2.5 is complete. 5. Proof of Theorem 2.6 Let (V˙ pk )0 ⊂ V˙ pk denote the subset of functions f satisfying the condition ⎛ ⎞ 1p ∑ lim sup ⎝ E k ( f ; Q) p ⎠ = 0; ε→0 d(π)≤ε

Q∈π

where d(π ) := sup Q∈π |Q|.

(5.1)

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We have to prove that if k > s := dp , then (V˙ pk )0 = V˙ kp . First, let us show that (V˙ pk )0 is a closed subspace of the space V˙ pk . To this end we define a seminorm T : V˙ pk → R+ given for f ∈ V˙ pk by ⎞ 1p

⎛ T ( f ) := lim sup ⎝ ε→0 d(π )≤ε

∑

E k ( f ; Q) p ⎠ .

Q∈π

Since T ( f ) ≤ | f |V pk for all f ∈ V˙ pk , seminorm T is continuous on V˙ pk . This implies that the preimage T −1 ({0}) = (V˙ pk )0 is a closed subspace of V˙ pk . Next, we show that V˙ kp is a closed subspace of (V˙ pk )0 . Since V˙ kp = clos(C ∞ ∩ V˙ pk , V˙ pk ) and (V˙ pk )0 is closed in V˙ pk , it suffices to prove that C ∞ ⊂ (V˙ pk )0 . To this end we estimate E k ( f ; Q), Q ∈ π, with f ∈ C ∞ by the Taylor formula as follows k

k

E k ( f ; Q) ≤ c(k, d)|Q| d max max |D α f | ≤ c(k, d, f )|Q| d . |α|=k

(5.2)

Q

This implies that ⎞ 1p

⎛ γ (π ; f ) ≤ c ⎝

∑

|Q|

k−s p+1 d

⎠ ≤ c max |Q| Q∈π

Q∈π

⎞ 1p

⎛ k−s d

∑ ⎝

(5.3)

|Q|⎠ ;

Q∈π

hereafter we set ⎞ 1p

⎛ γ (π ; f ) := ⎝

∑

E k ( f ; Q) p ⎠ .

(5.4)

Q∈π

Since s < k and the sum here ≤ 1, this implies that sup γ (π ; f ) ≤ cε

k−s d

→0

as ε → 0,

d(π )≤ε

i.e., f ∈ (V˙ pk )0 as required. Finally, let us prove that (V˙ pk )0 = V˙ kp . To this end we need the embedding (V˙ pk )0 ⊂ C

(5.5)

proved, in fact, in [9, Lm. 3.6]. Let us recall that as in this lemma the inequality ωk ( f ; t) ≤

sup osck ( f ; Q) ≤ 2d |Q|≤(kt)d

sup |Q|≤(kt)d

E k ( f ; Q) ≤ 2d

sup

γ (π ; f )

d(π)≤(kt)d

implies that for f ∈ (V˙ pk )0 lim ωk ( f ; t) = 0

t→0

while the Marchaud inequality, see, e.g., [10, Ch. 2, App. E2], implies from here that ω1 ( f ; t) → 0 as t → 0, i.e., f ∈ C.

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37

From (5.5) we obtain that (V˙ pk )0 ⊂ N V˙ pk , hence, due to Theorem 2.5 (c) the operator k L : V˙ pk → V˙ p∞ (sending a function of V˙ pk to its class of equivalence in L ∞ ) embeds (V˙ pk )0 k isometrically into V˙ p∞ . k Further, since L isometrically maps C ∞ as a subset of V˙ pk onto C ∞ as that of V˙ p∞ , it k k ∞ ˙k ˙ ˙ isometrically maps V p onto V p∞ (:= clos(C , V p∞ )). In turn, by [9, Thm. 2.13], k

˙ p∞ V

k = (V˙ p∞ )0 ,

(5.6)

where k (V˙ p∞ )0 :=

⎧ ⎪ ⎨

k f ∈ V˙ p∞

⎪ ⎩ E k∞ ( f ; Q) :=

⎞ 1p

⎫ ⎪ ⎬ ∑ p⎠ ⎝ =0 ; E k∞ ( f ; Q) : lim sup ⎪ ε→0 |π|≤ε ⎭ Q∈π ⎛

inf ∥ f − m∥ L ∞ (Q) .

d m∈Pk−1

k k Comparing definitions of (V˙ pk )0 (⊂ C) and (V˙ p∞ )0 we see that L((V˙ pk )0 ) ⊂ (V˙ p∞ )0 . k Since V˙ p ⊂ (V˙ pk )0 , the above implications yield k L(V˙ kp ) ⊂ L((V˙ pk )0 ) ⊂ (V˙ p∞ )0 = V˙ kp∞ = L(V˙ kp ).

Hence, L((V˙ pk )0 ) = L(V˙ kp ). This and injectivity of L|(V˙ pk )0 imply that (V˙ pk )0 = V˙ kp . The proof of the theorem is complete. 6. Proof of Theorem 2.8 The proof of the theorem is based on the following result of independent interest. Theorem 6.1. Let Q ⊊ Rd be a nontrivial closed cube and f ∈ ℓ∞ (Q). Then the following two-sided inequality with equivalence constants depending only on k, d holds: E k ( f ; Q) ≈ osck ( f ; Q).

(6.1)

Proof. We begin with the proof of the inequality E k ( f ; Q) ≤ c(k, d) osck ( f ; Q),

Q ⊂ Qd ,

(6.2)

where throughout the proof c(k, d) denotes a positive constant depending on k, d and changing from line to line or within a line. Without loss of generality it suffices to prove (6.2) for Q = Q d ; in this case, we write in (6.2) E k ( f ) and osck f omitting Q d , see Stipulation 1.10. For d = 1 the inequality was proved in [50]. In more details, let f ∈ ℓ∞ [0, 1] and L k f be i a polynomial of degree ≤ k − 1 interpolating f at points k−1 , i = 0, . . . , k, if k > 1 and equal 1 1 ( f (0) + f (1)) if k = 1. Clearly, L is a projection of ℓ [0, 1] onto Pk−1 . k ∞ 2 Now according to [50] ∥ f − L k f ∥∞ ≤ c osck f, where c = c(k).

(6.3)

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Further, let Qki be a linear subspace of ℓ∞ consisting of polynomials in xi of degree ≤ k − 1 with coefficients depending on the remaining variables x j ̸= xi . This is extended to k = 0 by setting Q d = {0}. Moreover, we define a projection of ℓ∞ onto Qki denoted by L ik given for f ∈ ℓ∞ by applying the interpolation operator L k to the function xi ↦→ f (x), 0 ≤ xi ≤ 1, with fixed variables x j ̸= xi . As a direct consequence of this definition and inequality (6.3) we have ∥ f − L ik f ∥∞ ≤ c oscik f,

(6.4)

where c = c(k) and { } oscik f := sup |∆kh f (x)| ,

(6.5)

x,h

where supremum is taken for the x, h ∈ Rd satisfying the condition (*) h is parallel to the xi -axis and x, x + kh ∈ Q d . Further, again directly from the definition we have the following: ′

If i ̸= i ′ , then projections L ik , L ik ′ commute.

Lemma 6.2.

Now let α ∈ Zd+ and L α :=

d ∏

L iαi .

(6.6)

i=1

Since the projections here pairwise commute, L α f is a polynomial in xi of degree ≤ αi − 1 for each 1 ≤ i ≤ d. Hence, L α f is a polynomial of vector degree β − e, where e := {1, . . . , 1} and βi ≤ αi , 1 ≤ i ≤ d. The linear space of such polynomials is denoted by Pα . Lemma 6.3.

There is a constant c = c(α, d) such that

∥ f − L α f ∥∞ ≤ c

d ∑

osciαi f.

(6.7)

i=1

Proof. Using the identity ⎞ ⎛ i−1 d ∑ ∏ ⎝ L αj ⎠ (1 − L iα ), 1 − L α = (1 − L 1α1 ) + i j i=2

j=1

the estimates 1 ≤ ∥L ik ∥ ≤ ∥L k ∥, 1 ≤ i ≤ d, and Lemma 6.2 we conclude that (d−1 ) d d ∏ ∑ ∑ i i ∥ f − L α f ∥∞ ≤ ∥L αi ∥ ∥ f − L αi f ∥∞ ≤ c(α, d) osciαi f. □ i=1

Corollary 6.4.

i=1

i=1

A function f ∈ ℓ∞ belongs to the space Pα if and only if

osciαi f = 0

for all

1 ≤ i ≤ d.

Proof. If (6.8) holds, then by (6.7) f = L α f ∈ Pα .

(6.8)

A. Brudnyi and Y. Brudnyi / Journal of Approximation Theory 251 (2020) 105346

39

α

Conversely, if f ∈ Pα , then ∆h i f = 0 for every h parallel to the xi axis, as f is a polynomial in xi of degree ≤ αi − 1, 1 ≤ i ≤ d. This and definition (6.5) imply (6.8). □ At the next step, we define a linear subspace of ℓ∞ by setting Qα :=

d ∑

Qαi i

(6.9)

i=1

and an operator acting in ℓ∞ by Lα := 1 −

d ∏ (1 − L iαi ).

(6.10)

i=1

Since 1 − L iαi annihilates Qαi i , 1 ≤ i ≤ d, and these operators pairwise commute, Lα is a ∏d projection of ℓ∞ onto Qα of norm ≤ i=1 (1 + ∥L iαi ∥) =: c(α, d). Now we estimate the order of approximation by Lα using mixed α-oscillation given for α ∈ Zd+ and f ∈ ℓ∞ by ) ⏐} {⏐( d ⏐ ⏐ ∏ ⏐ ⏐ αi (6.11) oscα f := sup ⏐ ∆h ei f (x)⏐ , i ⏐ ⏐ x,h i=1

where supremum is taken over x, h ∈ Rd satisfying the condition x ∈ Qd ,

xi + αi h i ∈ [0, 1], 1 ≤ i ≤ d,

i

(6.12) d

and {e }1≤i≤d is the standard orthonormal basis of R . In the sequel, we write ∆αh :=

d ∏

α

∆h i e i .

(6.13)

i

i=1 α

Here ∆h i ei = 1 if αi = 0; in particular, i

oscα f = oscik f Lemma 6.5.

if α = kei .

(6.14)

It is true that

∥ f − Lα f ∥∞ ≤ c(α, d) oscα f.

(6.15)

Proof (Induction on d). For d = 1 the assertion coincides with inequality (6.3). Now let (6.15) holds for d − 1 ≥ 0. Setting xˆ := (x1 , . . . , xd−1 ), αˆ := (α1 , . . . , αd−1 ), etc. ˆ for x ∈ Rd , α ∈ Zd+ and Qˆ := [0, 1]d−1 we then have for g ∈ ℓ∞ ( Q) ∥g − Lαˆ g∥ℓ∞ ( Q) ˆ d − 1) oscαˆ g. ˆ ≤ c(α, Now, for f ∈ ℓ∞ (Q d ), α ∈ Zd+ , we have from (6.10) (d−1 ) ∏ i f − Lα f = (1 − L αi ) ( f − L dαd f ) =: (1 − Lαˆ )ϕxd , i=1

where ϕxd : xˆ ↦→ ( f − L dαd f )(x, ˆ xd ), xˆ ∈ Qˆ d and xd ∈ [0, 1] is fixed.

(6.16)

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ˆ Taking here the ℓ∞ ( Q)-norm and applying (6.16) we obtain that ⏐} {⏐ ⏐ ⏐ ∥ f − Lα f ∥ℓ∞ ( Q) ˆ d − 1) oscαˆ (ϕxd ) = c(α, ˆ d − 1) sup ⏐(∆αhˆˆ (1 − L dαd ) f )(x)⏐ , ˆ ≤ c(α, x, ˆ hˆ

(6.17) where x, ˆ hˆ satisfy (6.12) for d − 1 (instead of d). Further, denoting the function xd ↦→ ∆αhˆˆ f (x), 0 ≤ xd ≤ 1, with fixed x, ˆ hˆ by ψx, ˆ hˆ , changing αˆ d the order of ∆hˆ and 1 − L αd and taking supremum over 0 ≤ xd ≤ 1 we obtain from (6.17) { } ∥ f − Lα f ∥∞ ≤ c(α, ˆ d − 1) sup ∥ψx, ˆ hˆ − L αd ψx, ˆ hˆ ∥ℓ∞ [0,1] . x, ˆ hˆ

Estimating the right-hand side by (6.3) we finally have ⏐} {⏐ ⏐ ⏐ α ∥ f − Lα f ∥∞ ≤ c(α, ˆ d − 1)c(αd ) sup ⏐∆αhˆˆ ∆h d ed f ⏐ =: c(α, d) oscα f. □ x,h

d

Now let ∏

Lk :=

Lα =

|α|=k

N ∏

Lα i ,

(6.18)

i=1

where N is the cardinality of the set {α ∈ Zd+ : |α| = k} and {α i }1≤i≤N is its arbitrary enumeration. Lemma 6.6.

It is true that

∥ f − Lk f ∥∞ ≤ c(k, d)

∑

oscα f.

(6.19)

|α|=k

Proof. Using the identity ⎛ ⎞ N i−1 ∑ ∏ ⎝ Lα j ⎠ (1 − Lαi ), 1 − Lk = i=1

where

j=1

∏i−1

:= 1 for i = 1 and applying Lemma 6.5 we have (recall that all ∥Lαi ∥ ≥ 1) ( N −1 ) N ∏ ∑ ∑ ∥ f − Lk f ∥∞ ≤ ∥Lαi ∥ c(α i , d) oscαi f ≤ c(k, d) oscα f. □ j=1

i=1

Lemma 6.7.

i=1

|α|=k

E k ( f ) ≤ ∥ f − Lk f ∥∞ .

Proof. It suffices to prove that Lk f is a polynomial of degree ≤ k − 1. To this end we note that Lα f = f if oscα f = 0

for all

|α| = k,

(6.20)

see (6.19). Since oscik f = osckei f , see (6.14), and the latter satisfies (6.20) for all 1 ≤ i ≤ d, Corollary 6.4 implies that f is a polynomial of degree ≤ k − 1 in each variable.

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Due to (6.20) for every |α| = k the mixed difference ∆αh f = 0. Dividing by h α and sending h to 0 we obtain that D α f = 0, |α| = k. Since for every monomial x β with |β| ≥ k there is |α| = k such that D α (x β ) ̸= 0, the polynomial f , hence, Lα f , contains in its decomposition only monomials x β with |β| ≤ k − 1. d Hence, Lα f ∈ Pk−1 . □ It remains to use Corollary E.4 of [10, Ch. 2, App. E] which, in particular, implies that oscα f ≤ c(k, d) osck f,

(6.21)

|α| = k.

The corollary is formulated for continuous f but its proof is based only on the combinatorial identity relating ∆αh with a linear combination of shifted k-differences, see Theorem E.1 there. Hence, inequalities (6.21) hold for f ∈ ℓ∞ as well. Finally, we combine Lemma 6.6, 6.7 and inequalities (6.21) to obtain the required inequality E k ( f ) ≤ c(k, d) osck f, see (6.2). The proof of the converse inequality is essentially simpler. In fact, since by the definition, d see (1.4), osck f ≤ 2k ∥ f ∥∞ and osck |P d = 0, for every polynomial m ∈ Pk−1 k−1

osck f ≤ osck ( f − m) ≤ 2 ∥ f − m∥∞ . k

Taking here infimum in m we get 2−k osck f ≤ E k ( f ). The proof of Theorem 6.1 is complete. □ Now Theorem 2.8 follows directly from Theorem 6.1 and the definition of | · |V pk , see (1.20). 7. Proof of Theorem 2.11 (a) Let f ∈ V˙ pk and s := that

d p

∈ (0, k). Given ε > 0 we have to find a function f ε ∈ Λk,s such

|{x ∈ Q d : f (x) ̸= f ε (x)}| < ε.

(7.1) d

To this end we first find a set, say, S f ⊂ Q of measure 1 such that f is locally Lipschitz at every its point. Then we find a subset of S f denoted by Sε such that its Lebesgue measure is at least 1 − ε and the trace f | Sε satisfies the conditions of the extension theorem for Lipschitz functions. Extending f | Sε to a function of Λk,s we finally obtain the required f ε of (7.1). We begin with a result on the structure of E k ( f ; Q) as a function of Q ⊂ Q d . Parameterizing the set of cubes in Rd by the bijection Rd × R+ ∋ (x, r ) ↔ Q r (x) ⊂ Rd , where Q r (x) is a closed cube of sidelength 2r > 0 centered at x, we consider E k ( f ; Q) as a function of x, r . It is easily seen that the subset Ω0 ⊂ Rd × R+ of cubes Q r (x) ⊂ Q d is a convex body. Proposition 7.1. Let f ∈ V˙ pk , s ∈ (0, k]. There is a function Ek ( f ; ·) : Ω0 → R+ such that the following is true. (a) Ek ( f ; ·) is Lebesgue measurable in x ∈ Q d . (b) For every Q ⊂ Q d E k ( f ; Q) ≤ Ek ( f ; Q).

(7.2)

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(c) For every packing π ∈ Π ⎧ ⎫1 ⎨∑ ⎬p p ≤ 5| f |V pk . Ek ( f ; Q) ⎩ ⎭

(7.3)

Q∈π

Proof. Let { f n }n∈N ⊂ C ∞ be a sequence converging pointwise on Q d to f such that lim | f n |V pk ≤ 5| f |V pk ,

n→∞

lim ∥ f n ∥∞ ≤ 3∥ f ∥∞ ,

(7.4)

n→∞

see Theorem 2.3. Then we define the required function by setting for Q ⊂ Q d Ek ( f ; Q) := lim E k ( f n ; Q)

(7.5)

n→∞

and prove that Ek ( f ; ·) satisfies the declared properties. (a) First we show that E k (g; ·) is continuous in x ∈ Q d if g is. Let Q ⊂ Q d and h ∈ Rd be such that Q + h ⊂ Q d . Then we have |E k (g; Q) − E k (g; Q + h)| = |E k (g; Q) − E k (g(· − h); Q)| ≤ E k (g − g(· − h); Q) ≤ max |g(x) − g(x − h)| → 0 Q

as

h → 0.

In other words, lim E k (g; x + h, r ) = E k (g; x, r )

h→0

at every x from the set Qr := {x ∈ Q d : Q r (x) ⊂ Q d },

(7.6)

i.e., E k (g; ·) is continuous on the compact convex set Qr . Now since each f n ∈ C ∞ approximating f is continuous on Q d , the function E k ( f n ; ·; r ) is continuous in x, hence, Ek ( f ; ·; r ) is Lebesgue measurable in x ∈ Qr , see (7.5). Therefore it is Lebesgue measurable in x on the set ∪r >0 Qr = Q˚ d . (b) To prove that E k ( f ; Q) ≤ Ek ( f ; Q), Q ⊂ Q d , for every n ∈ N we choose a polynomial d Pn ∈ Pk−1 such that E k ( f n ; Q) = ∥ f n − Pn ∥ℓ∞ (Q) . We have ∥Pn ∥ℓ∞ (Q) ≤ ∥ f n ∥∞ + E k ( f n ; Q) ≤ 2∥ f n ∥∞ and then by (7.4) lim ∥Pn ∥ℓ∞ (Q) ≤ 2 lim ∥ f n ∥∞ ≤ 6∥ f ∥∞ .

n→∞

n→∞

Hence, {Pn }n∈N is bounded in ℓ∞ (Q) and so contains a subsequence {Pni }i∈N uniformly d converging on Q to some P ∈ Pk−1 such that lim ∥ f ni − Pni ∥ℓ∞ (Q) = lim ∥ f n − Pn ∥ℓ∞ (Q) .

i→∞

n→∞

This then implies that E k ( f ; Q) ≤ ∥ f − P∥ℓ∞ (Q) ≤ lim ∥ f ni − Pni ∥ℓ∞ (Q) = lim ∥ f n − Pn ∥ℓ∞ (Q) =: Ek ( f ; Q). i→∞

n→∞

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43

This proves (7.2) and property (b). (c) To prove (7.3) we write ∑ ∑ ∑ Ek ( f ; Q) p := lim E k ( f n ; Q) p ≤ lim E k ( f n ; Q) p Q∈π

≤

Q∈π n→∞ p lim | f n |V k p n→∞

n→∞ Q∈π

≤ (5| f |V pk ) . p

This proves (7.3) and the proposition. □ Proposition 7.2. Under the assumptions of Proposition 7.1 ) ( 1 lim |Q|− p E k ( f ; Q) < ∞ Q→x

(7.7)

for almost all x ∈ Q d . Proof. Let ρa : Qa → R+ , a > 0, be a function given by { d } ρa (x) := sup r − p Ek ( f ; x, r ) : r ≤ a ;

(7.8)

d

here Qa is the subset of Q given by (7.6). Given N ∈ N we then set Sa,N := {x ∈ Qa : ρa (x) > N }. Lemma 7.3.

(7.9)

Let | f |V pk = 1. Then

|Sa,N | ≤ c(d, p)N − p .

(7.10)

Proof. Let S ⊂ Q d and ⎧ ⎫ ⎨∑ ⎬ M(S) := inf |Q| , ⎭ ∆ ⎩

(7.11)

Q∈∆

where ∆ runs over all coverings of S by subcubes of Q d . It is known that for a measurable set S (7.12)

|S| ≤ M(S),

see, e.g., [16, Thm. I.1]. Now let x ∈ Sa,N . By definitions (7.8), (7.9) there is a cube Q x of radius ≤ a centered at x ∈ Qa such that 1

Ek ( f ; Q x ) > N |Q x | p .

(7.13)

Since the family {Q x } covers Sa,N by centers of its cubes, the Besicovich covering theorem, see, e.g., [20, Thm. 1.2], asserts that there is a countable subcovering of {Q x }, say {Q i }i∈N , which is the union of at most c(d) packings πi ∈ Π . This and (7.11)–(7.13) imply that ∑ ∑ ∑ ∑ |Sa,N | ≤ |Q| ≤ N − p Ek ( f ; Q) p . i≤c(d) Q∈πi

i≤c(d) Q∈πi

By (7.3) the inner sum in the right-hand side is bounded from above by (5| f |V pk ) p = 5 p , hence, |Sa,N | ≤ 5 p c(d)N − p . □

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Now we have from (7.10) ⏐ ⏐ ⏐ ⏐⋂ ⏐ ⏐ Sa,N ⏐ = 0; |Sa,∞ | = ⏐ ⏐ ⏐ N ∈N

moreover, by monotonicity of sets Sa,∞ under inclusions, S∞ := ∪a>0 Sa,∞ is of (Lebesgue) measure zero. Finally, for every x ∈ Qa \ Sa,∞ ⋃ d sup r − p Ek ( f ; x, r ) < ∞ and Qa = Q˚ d , (7.14) r ≤a

0
see (7.6). Hence, for every point x ∈ Q d outside the set S∞ ∪ ∂ Q d of measure 0 1

lim |Q|− p Ek ( f ; Q) < ∞.

Q→x

This and (7.2) give (7.7). Proposition 7.2 is proved. □ In the next auxiliary result we use the following: Definition 7.4. A Lebesgue measurable set S ⊂ Rd is said to be Ahlfors d-regular (briefly, regular) if for every x ∈ S and some constants δ ∈ (0, 1), r0 > 0, |Q r (x) ∩ S| ≥ δ|Q r (x)|,

0 < r ≤ r0 .

Lemma 7.5. Let f ∈ V pk , s := Σε ⊂ Q d such that

d p

(7.15)

∈ (0, k]. Given ε ∈ (0, 1) there is a regular set, say,

|Σε | ≥ 1 − ε

(7.16)

and, moreover, there are positive constants cε , aε such that sup E k ( f ; x, r ) ≤ cε r s ,

0 ≤ r ≤ aε .

(7.17)

x∈Σε

Proof. We set √ ( ) ε 1 aε := 1− d 1− . 2 3

(7.18)

By (7.14) we have for every x ∈ Qaε \ Saε ,∞ and rn := 2−n aε ϕ(x) := sup {rn−s Ek ( f ; x, rn )} < ∞.

(7.19)

n∈Z+

Since the sequence under supremum is measurable in x, see Proposition 7.1 (a), ϕ is measurable and finite at every point of Qaε \ Saε ,∞ . Hence, there are a set Ωε ⊂ Qaε \ Sa,∞ and a constant Nε such that ε ϕ ≤ Nε on Ωε and |Ωε | ≥ |Qaε | − . (7.20) 3 Further, the Lebesgue density theorem implies that |Ωε ∩ Q r (x)| lim = 1 a.e. in Ωε . (7.21) r →0 |Q r (x)|

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Then by Egorov’s theorem there is a measurable subset Σε ⊂ Ωε of measure ε |Σε | > |Ωε | − 3 } { ∩Q r (·)| converges to 1 uniformly on Σε as r → 0. such that the family of functions |Ωε(2r d ) r ∈(0,aε ] Assuming without loss of generality that all x ∈ Σε are simultaneously density points of Ωε and Σε we also have |Ωε ∩ Q r (x)| |Σε ∩ Q r (x)| 1 = lim = lim r →0 r →0 (2r )d (2r )d for every x ∈ Σε . { } ∩Q r (·)| This implies that the family of functions |Σε(2r also converges to 1 uniformly )d r ∈(0,aε ] on Σε as r → 0. Hence, for every x ∈ Σε and some 0 < r0 < 1 (depending on Ωε ) |Σε ∩ Q r (x)| 1 > for 0 < r ≤ r0 , d (2r ) 2 i.e., Σε is regular. Let us show that Σε satisfies inequality (7.16). In fact, due to (7.6) and (7.18) 2ε 2ε ε = (1 − 2aε )d − = 1 − ε, |Σε | > |Ωε | − > |Qaε | − 3 3 3 as required. Finally, ϕ ≤ Nε on the regular set Σε , hence, for x ∈ Σε Ek ( f ; x, 2−n aε ) ≤ Nε (2−n aε )s ,

n ∈ Z+ .

Enlarging Nε (and denoting its new value by cε ) we can replace here 2−n aε by an arbitrary r ∈ (0, aε ]. Moreover, by Proposition 7.1 (b), E k ( f ; ·) ≤ Ek ( f ; ·), hence, for x ∈ Σε E k ( f ; x, r ) ≤ cε r s ,

0 < r ≤ aε .

This gives (7.17) and proves the lemma. □ Proceeding the proof of Theorem we apply inequality (7.17) to the trace f |Σε . This function satisfies the inequality E k ( f |Σε ; Q r (x)) ≤ cε r s ,

0 < r ≤ aε ,

(7.22)

on the regular set Σε and therefore meets the conditions of the extension theorem [5], see [11, Thm. 9.30] for the general version of this result. Due to this theorem there is a function f ε : Rd → R such that f ε |Σε = f |Σε and, moreover, sup E k ( f ε ; x, r ) ≤ cr s ,

0 ≤ r ≤ 1;

x∈Q d

here c depends on ε, d, k and f . Using further Theorem 6.1 we replace E k ( f ε ; x, r ) by osck ( f ; Q r (x)) and finally obtain |∆kh f ε (x)| ≤ osck ( f ; x, k∥h∥) ≤ c∥h∥s , where ∥h∥ := max1≤i≤d |h i | and x, x + h belong to Q d .

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By definition, see (2.8), (2.9), this means that f ε ∈ Λk,s , 0 < s ≤ k. Thus, we have proved that for f ∈ V˙ pk with s ∈ (0, k] and ε > 0 there is a Lipschitz function f ε ∈ Λk,s such that f = fε

outside a set of Lebesgue measure < ε.

(7.23)

For 0 < s < k, this establishes assertion (a) of Theorem 2.11. Finally, in part (b) of this theorem we have to prove that if s = k we can replace the function f ε ∈ Λk,k by a C k function. To this end we use the linear continuous isomorphism k Λk,k = C k−1,1 ∼ . = W˙ ∞

(7.24)

see Proposition 2.10 and [5], respectively. 1 More precisely, every class f ∈ W˙ ∞ contains a unique C k−1,1 function, say fˆ, such that k the map f ↦→ fˆ is a linear continuous bijection of W˙ ∞ onto C k−1,1 . k k Further, W˙ ∞ ⊂ W˙ p , p > 1, while for every function fˆ from the class f ∈ W˙ pk there is a function f ε ∈ C k such that f coincides with f ε outside a set of measure < ε, see [15, Thm. 4.13]. Combining this with (7.24) we conclude that for every g ∈ C k−1,1 there is gε ∈ C k coinciding with g outside of a set of measure < ε. Taking now f ∈ V˙ pk with s = k and g = f ε from (7.23) we obtain the function gε ∈ C k coinciding with f outside a set of measure < 2ε. This proves assertion (b) and the theorem. 8. Proof of Theorem 2.15 (a) We have to prove that if f ∈ V˙ pk and s := f ∈ Λk,s (x)

d p

∈ (0, k], then

a. e.

(8.1)

To this end we use Proposition 7.2 asserting that for this f lim r −s E k ( f ; x, r ) < ∞ for all

r →0

x ∈ Sf ,

(8.2)

where S f is a subset of Q˚ d of measure 1. Due to inequality (2.7) E k ( f ; ·) here can be replaced by osck ( f ; ·); after this change condition (8.2) at x ∈ S f coincides with the condition of Definition 2.12 introducing the space Λk,s (x0 ). Hence, we obtain that f ∈ Λk,s (x) for all x ∈ S f ⊂ Q˚ d , i.e., almost everywhere on Qd . (b) We have to prove that the function under consideration belongs to the Taylor space T s (x) for almost all x ∈ Q d . To this end we use Theorem 3 from [7, §2] that, in particular, asserts the following: If a function f ∈ ℓ∞ ( Q˚ d ) satisfies at a given point x0 the condition lim r −s E k ( f ; x, r ) < ∞,

r →0

where 0 < s ≤ k, then f ∈ T s (x0 ). It follows from here and condition (8.2) that f ∈ T s (x) for every x ∈ S f , i.e., for almost all x ∈ Q d . (c) Now we have to show that if f ∈ V˙ pk and s = k, then f ∈ t k (x0 ) a.e. To this end, first we note that due to the previous result f ∈ T k (x) for every x ∈ S f , d i.e., for every such x there are the Taylor polynomial Tx ( f ) ∈ Pk−1 and constants c1 , c2 > 0

A. Brudnyi and Y. Brudnyi / Journal of Approximation Theory 251 (2020) 105346

47

independent of r such that max | f − Tx ( f )| ≤ c1r k Q r (x)

0 < r ≤ c2 .

for

(8.3)

Further, we use Theorem 2.11 asserting that given ε > 0 there are a regular subset Sε ⊂ Q˚ d and a function f ε ∈ C k such that |Sε | > 1 − ε

and

on

f = fε

Sε .

(8.4)

Now we derive from here and (8.3) that f ∈ t k (x)

x ∈ Sε .

for almost all

(8.5) k

d

Since | ∪ε>0 Sε | = 1, this implies that f ∈ t (x) for almost all x in Q . Let Tx ( f ε ) ∈ Pkd be the Taylor polynomial of f ε at x, i.e., max | f ε − Tx ( f ε )| ≤ c(r )r k , Q r (x)

0
1 , 2

(8.6)

where c(r ) → 0 as r → 0. We write ∑ Tx ( f ε ) := Tˆx ( f ε ) + cα ( f ε )(· − x0 )α , |α|=k

where Tˆx ( f ε ) is the Taylor polynomial for f ε at x of degree k − 1. In particular, 1 max | f ε − Tˆx ( f ε )| ≤ c1r k for 0 < r ≤ Q r (x) 2 with a constant independent of r . Lemma 8.1.

(8.7)

If x ∈ Sε ∩ S f , then

Tx ( f ) = Tˆx ( f ε ).

(8.8)

Proof. By regularity of Sε there are positive constants γ , r0 such that for every x ∈ Sε |Sε ∩ Q r (x)| ≥ γ |Q r (x)|

for

0 < r ≤ r0 .

(8.9)

Moreover, f = f ε on Sε ∩ Q r (x); hence, max |Tx ( f ) − Tˆx ( f ε )| ≤ max | f − Tx ( f )| + max | f ε − Tˆx ( f ε )|.

Sε ∩Q r (x)

Q r (x)

Q r (x)

Estimating the right-hand side by inequalities (8.3) and (8.7) we have for every x ∈ Sε ∩ S f max |Tx ( f ) − Tˆx ( f ε )| ≤ c3r k ,

Sε ∩Q r (x)

0 ≤ r ≤ c2 ,

(8.10)

for some c3 > 0 independent of r . Further, using a Remez type inequality [12] and (8.9) we can estimate the left-hand side of (8.10) from below by c(k, d)γ k−1 max Qr (x) |Tx ( f ) − Tˆx ( f ε )|. This implies for sufficiently small r > 0 the inequality max |Tx ( f ) − Tˆx ( f ε )| ≤ cr k Q r (x)

with c independent of r . Dividing this by r k and letting r to 0 we conclude that every coefficient of this polynomial (of degree ≤ k − 1) should be 0. □

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Finally, we show that f ∈ t k (x) for almost all x ∈ Sε . In fact, for the function F := f − f ε and x ∈ Sε ∩ S f we have by Lemma 8.1 and inequalities (8.3), (8.7) max |F| ≤ max | f ε − Tˆx ( f ε )| + max | f − Tx ( f )| ≤ c1r k , 0 < r ≤ c2 , (8.11) Q r (x)

Q r (x)

Q r (x)

where c1 , c2 > 0 are independent of r . Inequality (8.11) and Theorem 10 from [15] imply that for almost all x ∈ Sε ∩ S f max |F| = o(r k ),

r → 0.

Q r (x)

In turn, for such x we derive from here and (8.6) max | f − Tx ( f ε )| ≤ max |F| + max | f ε − Tx ( f ε )| = o(r k ), Q r (x)

Q r (x)

Q r (x)

r → 0.

Since |Sε ∩ S f | = |Sε |, f ∈ t k (x) for almost all x ∈ Sε . This proves assertion (c) of Theorem 2.15. 9. Proofs of Theorem 2.18 and Corollary 2.20 Proof of Theorem 2.18. (a) We have to show that the map L : f ↦→ f˜ sending a function f ∈ N V˙ pk to its equivalence class f˜ ∈ L p is a linear continuous injection of N V˙ pk in Λks p∞ , 0 < s ≤ k. In fact, injectivity of the map follows from the definition of the class N V˙ pk , see Corollary 2.2, and so it remains to show that its image belongs to Λks p∞ and the map to this space is continuous. To this end, we apply again Theorem 4 from [6, §2] to write ⎞ 1p ⎛ ∑ E kp ( f˜; Q) p ⎠ , t −s ωkp ( f˜; t) ≤ c t −s ⎝ Q∈π

where π is some packing in Π (Q d ) containing cubes of volume ≤ t d . For these cubes, we clearly have 1

d

E kp ( f˜; Q) ≤ |Q| p E k∞ ( f˜; Q) ≤ c t p E k ( f ; Q) := c t s E k ( f ; Q). Together with the previous inequality this implies ⎛ ⎞ 1p ∑ t −s ωkp ( f˜; t) ≤ c ⎝ E k ( f ; Q) p ⎠ . Q∈π

Taking here supremums over t > 0 and all π ∈ Π , we finally have ⎛ ⎞ 1p ∑ | f˜|Λksp∞ := sup t −s ωkp ( f˜; t) ≤ c sup ⎝ E k ( f ; Q) p ⎠ =: c | f |V pk . t>0

π

Q∈π

Hence, the map under consideration acts continuously from N V˙ pk in Λks p∞ , i.e., gives the , see (2.24). required implication N V˙ pk ↪→ Λks p∞ Further, let the space V˙ pk be such that s :=

d ∈ (0, k). p

(9.1)

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ˆ We will show that every class f ∈ Λks p1 contains a unique representative f ∈ C and the ks k map f ↦→ fˆ is a linear continuous injection of Λ p1 into V˙ p . This clearly proves the required implication ˙k Λks p1 ↪→ V p .

(9.2)

This will complete the proof of part (a) of Theorem 2.18. To prove (9.2), given f ∈ L p we denote by L f the Lebesgue set of f , i.e., Q d \ L f is of measure zero and the limit over cubes Q centered at x ∫ 1 f dy (9.3) fˆ(x) := lim Q→x |Q| Q exists at every x ∈ L f . Lemma 9.1. If f ∈ L p is such that ∫ 1 ωkp ( f ; t) dt < ∞ t s+1 0

(9.4)

for 0 < s < k, then fˆ is uniformly continuous on L f and for every subcube Q ⊂ Q d 1

∫ E k∞ ( f ; Q) ≤ c(k, d) 0

|Q| d

ωkp ( f ; t; Q) dt. t s+1

(9.5)

Hereafter, as above we set d E kp ( f ; Q) := inf{∥ f − m∥ L p (Q) : m ∈ Pk−1 }

(9.6)

and denote by m Q ( f ) an optimal polynomial for this relation. Proof. Setting for brevity ω(t) := ωkp ( f ; t; Q),

t > 0,

we estimate the nonincreasing rearrangement ( f − m Q ( f ))∗ by 1 ∫ |Q| ω(u d ) ∗ du, 0 ≤ t ≤ |Q|, ( f − m Q ( f )) (t) ≤ c(k, d) 1 t u 1+ p 2 see [7, App. II, Cor. 2′ ]. Sending here t to 0 and noting that s = dp we then obtain

(9.7)

1

∗

|Q| d

∫

E k∞ ( f ; Q) ≤ ∥ f − m Q ( f )∥ L ∞ (Q) = lim ( f − m Q ( f )) (t) ≤ c(k, d) t→∞

0

ω(u) du. u s+1

This proves (9.5). Now we show that the function fˆ : L f → R is uniformly continuous. In fact, by (9.5) lim E k∞ ( f ; Q) = 0

Q→x

(9.8)

and the convergence is uniform in x. As in the proof of Theorem 2.6 we derive from here that lim ω1∞ ( f ; t) = 0.

t→0

(9.9)

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Finally, let x, x + h ∈ L f (⊂ Q˚ d ). Then ∫ 1 | f (y + h) − f (y)| dy | fˆ(x + h) − fˆ(x)| ≤ lim Q→x |Q| Q ≤ ω1∞ ( f ; ∥h∥) → 0 as h → 0. Hence, fˆ is uniformly continuous on the dense in Q d set L f and so can be continuously extended to Q d ; we preserve the same notation fˆ for the extension. Lemma 9.1 is proved. □ Now let us check that the regularization fˆ, see (9.3), satisfies E k∞ ( f ; Q) = E k ( fˆ; Q) Indeed, let m Q ∈

d Pk−1

for all

Q ⊂ Qd .

(9.10)

be such that

E k∞ ( f ; Q) = ∥ f − m Q ∥ L ∞ (Q) . Given ε ∈ (0, 1) we choose fˆε ∈ ℓ∞ of the class f ∈ L ∞ such that sup | fˆε − m Q | ≤ ∥ f − m Q ∥ L ∞ (Q) + ε. Q

Since fˆ and fˆε coincide on a set of complete Lebesgue measure, say Sε , we have fˆ − m Q = fˆε − m Q

on

Sε ∩ L f .

This, in turn, implies E k ( fˆ; Q) ≤ sup | fˆ − m Q | − sup | fˆ − m Q | ≤ sup | fˆε − m Q | Q

Q

Sε ∩L f

≤ ∥ f − m Q ∥ L ∞ (Q) + ε = E k∞ ( f ; Q) + ε. Hence, E k ( fˆ; Q) ≤ E k∞ ( f ; Q). Since the converse inequality is evident, (9.10) is proved. Now let π ∈ Π be a packing. Due to Lemma 9.1, equality (9.10) and Minkowski inequality we have ⎛ ⎞ 1p ⎛ ⎛ ⎞ p ⎞ 1p 1 ∑ ∑ ∫ |Q| d ωkp ( f ; t; Q) ⎝ ⎝ E k ( fˆ; Q) p ⎠ ≤ c ⎝ dt ⎠ ⎠ s+1 t 0 Q∈π Q∈π (9.11) ∫ ≤c 0

1

⎞ 1p

⎛ t −s−1 ⎝

∑

ωkp ( f ; t; Q) p ⎠

dt.

Q∈π

Hereafter all constants depend only on k, d. Lemma 9.2. It is true that ⎞ 1p ⎛ ∑ ⎝ ωkp ( f ; t; Q) p ⎠ ≤ c ωkp ( f ; t). Q∈π

(9.12)

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Proof. By Theorem 4 from [6, §2] ( ) 1p ∑ p ωkp ( f ; t; Q) ≈ sup E kp ( f ; R) , π ∈Πt (Q)

51

(9.13)

R∈π

where Πt (Q) consists of all packings in Q with cubes of volume ≤ t d . Hence, for every Q there is a packing, say π Q ∈ Πt (Q), such that ⎛ ⎞ 1p ∑ E kp ( f ; R) p ⎠ . ωkp ( f ; t; Q) ≤ c ⎝ R∈π Q

(∑ )1 ∑ p p Therefore the left-hand side of (9.12) is bounded by c . Q∈π R∈π Q E kp ( f ; R) d Since the set of subcubes ∪ Q∈π π Q is a packing in Q containing only cubes of volume ≤ t d , equivalence (9.13) bounds the above expression by c ωkp ( f ; t) (= c ωkp ( f ; t; Q d )). This proves (9.12). □ Now combining (9.11) and (9.12) and taking supremum over π ∈ Π we finally have ⎞ 1p ⎛ ∫ 1 ∑ ωkp ( f ; t) | fˆ|V pk := sup ⎝ E k ( fˆ; Q) p ⎠ ≤ c dt =: c| f |Λks . p1 t s+1 π 0 Q∈π ˙k ˙k This proves that the linear injection f ↦→ fˆ maps Λks p1 in V p ∩ C ⊂ N V p and is bounded, i.e., implication (9.2) holds. Theorem 2.18 (a) is proved. (b) Now we have to prove that if k = s = d, then the following linear isomorphism is given by the operator L| N V˙ pk Li p1d ∼ = N V˙1d .

(9.14)

Since injectivity of L| N V˙ pk : N V˙ pk → Λks p∞ , 0 < s ≤ k, of part (a) of the theorem implies for k = s = d injectivity of L| N V˙ d : N V˙1d → Li p1d , it remains to prove that it is onto. In the 1 derivation, we use the following Sobolev type embedding that is interesting in its own right. Proposition 9.3.

It is true that

d Li p1d ⊂ V˙1∞ .

(9.15)

Proof. In the proof, we use the following known results. Let BV k (Q) be the linear space of L 1 functions on Q whose kth distributional derivatives are finite Borel measures. It is endowed by a seminorm given for f ∈ L 1 (Q) by { } | f | BV k := max var D α f . (9.16) |α|=k

Theorem A ([5] Theorem 4). It is true that Li p1k (Q) = BV k (Q) with equivalence constants of seminorms depending only on k and d. The second result is the embedding theorem for Sobolev space W1d := W1d (Q d ).

(9.17)

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Theorem B ([23]). It is true that W1d ↪→ C,

(9.18)

where the embedding constant depends only on d. As a consequence we have the following: Lemma 9.4.

For each function f ∈ BV d (Q), Q ⊂ Q d , it is true that f ∈ L ∞ and

E d∞ ( f ; Q) ≤ c(d)| f | BV d (Q) . Proof. We begin with some facts of interpolation space theory. First, we recall that the K -functional of an embedded pair X ⊂ Y of Banach spaces is a function on Y × R+ given for y ∈ Y , t ∈ R+ by { } K (t; y; X, Y ) := inf ∥x∥ X + t∥z∥Y . (9.19) y=x+z

Second, the relative (Gagliardo) completion of X in Y denoted by X c,Y is a Banach space whose closed unit ball is the closure of that of X in Y . The relation between these notions is given by the following: Proposition 9.5 ([13], Prop. 2.2.20). It is true that ∥y∥ X c,Y = sup t −1 K (t; y; X, Y ).

(9.20)

t>0

Further, we use the following well-known result, see, e.g., [28] and references therein, 1

t > 0,

K (t; f ; L p ; W pk ) ≈ t∥ f ∥ p + ωkp ( f ; t k ),

(9.21)

with constants of equivalence depending only on k and d. Along with (9.20) and Theorem A this implies ∥ f ∥(W k )c,L 1 ≈ ∥ f ∥1 + sup 1

t>0

ωk1 ( f ; t) := ∥ f ∥1 + | f | Li pk ≈ ∥ f ∥1 + | f | BV k . 1 tk

This and Theorem B further imply ( ) ∥ f ∥C c,L 1 ≤ c(d) ∥ f ∥1 + | f | Li pd .

(9.22)

(9.23)

1

By definition the closed unit ball of C c,L 1 is the closure of the closed unit ball of C in L 1 , hence, (C c,L 1 , ∥ · ∥C c,L 1 ) = (L ∞ , ∥ · ∥∞ ). Using this and the homothety of Rd mapping Q d onto Q we transform (9.23) to the inequality ( ) ∥ f ∥ L ∞ (Q) ≤ c(d) |Q|−1 ∥ f ∥ L 1 (Q) + | f | Li pd (Q) . 1

d Applying this to the function f − m Q , where m Q ∈ Pk−1 is optimal for the distance from f d to Pk−1 in L 1 (Q), we have ( ) E d∞ ( f ; Q) ≤ ∥ f − m Q ∥ L ∞ (Q) ≤ c(d) |Q|−1 E d1 ( f ; Q) + | f | Li pd (Q) . (9.24) 1

Finally, by the Taylor formula for functions g ∈ W1d (Q) we have E d1 (g; Q) ≤ c(d) |Q| · |g|W d (Q) . 1

(9.25)

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53

Since due to (9.22) the Banach space (BV d (Q), ∥ · ∥ L 1 (Q) + | · | BV d (Q) ) coincides (up to equivalence of norms) with the relative completion of W1d (Q) in L 1 (Q), for every f ∈ BV d (Q) = Li p1d (Q) (see Theorem A) there is a sequence { f n }n∈N ⊂ W1d (Q) such that f n → f in L 1 (Q) and | f n |W d (Q) → | f | BV d (Q) as n → ∞. 1 Applying (9.25) to every f n , then passing to limit and using Theorem A we obtain from (9.24) the required inequality ( ) E d∞ ( f ; Q) ≤ c(d) |Q|−1 · |Q| · | f | BV d (Q) + | f | Li pd (Q) ≤ c1 (d) | f | BV d (Q) . (9.26) 1

Lemma 9.4 is proved. □ Now let π be a packing in Q d . Using Lemma 9.4 for each Q ∈ π and f ∈ BV d we have ⎧ ⎫ ⎨∑ ⎬ ∑ ∑ E d∞ ( f ; Q) ≤ c(d) | f | BV d (Q) ≤ c(d) max var Q D α f ⎭ |α|=d ⎩ Q∈π

Q∈π

Q∈π

≤ c(d) max var Q d D α f := c(d)| f | BV d . |α|=d

Taking here supremum over π we then obtain by Theorem A | f |V d ≤ c(d)| f | BV d ≤ c1 (d)| f | Li pd . 1∞

1

In other words, we proved the embedding d Li p1d ⊂ V˙1∞ .

(9.27)

This completes the proof of Proposition 9.3 □ To complete the proof of part (b) of the theorem, we consider the composition of continuous d embeddings i 1 : N V˙1d ↪→ Li p1d , see (2.24), and i 2 : Li p1d ⊂ V˙1∞ . The resulting embedding d i 2 ◦ i 1 coincides with L| N V˙ d , i.e., sends a function in N V˙1 ⊂ ℓ∞ to its equivalence class in 1 d d V˙1∞ ⊂ L ∞ . But according to Theorem 2.5 (c) it maps N V˙1d isometrically onto V˙1∞ . Hence, d d d ∼ d ˙ ˙ Li p1 = V1∞ and N V1 = Li p1 . The proof of Theorem 2.18 is complete. □ Proof of Corollary 2.20. First, we prove that each f˚ ∈ W˙ 1d can be represented by a function f ∈ AC (i.e., f ∈ f˚) such that | f |V d ≤ c| f˚|W d 1

(9.28)

1

for some c depending only on d. To this end we use the Gagliardo theorem [23] asserting that each g˚ ∈ W˙ 1d (Q), Q ⊂ Q d , can be represented by a (unique) continuous function g and there is a linear projection d PQ : L 1 (Q) → Pd−1 such that ( ) ˚ + |g| ˚ W d (Q) . max |g| ≤ c(d) max |PQ (g)| (9.29) Q

Q

1

Let f ∈ C be the representative of f˚ (∈ W˙ 1d ). Applying (9.29) to g˚ = f˚| Q − PQ ( f˚| Q ) and to its continuous representative g := f | Q − PQ ( f˚| Q ) we obtain (as PQ (g) = 0): E d ( f ; Q) ≤ max | f − PQ ( f˚)| ≤ c(d)| f˚|W d (Q) . Q

1

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Then for every packing π ∈ Π ∑ ∑ ∫ E d ( f ; Q) ≤ c(d) Q∈π

|α|=d

|D α f | d x,

(9.30)

∪π

where ∪ π := ∪ Q. Q∈π

By absolute continuity of the Lebesgue integral the right-hand side here tends to 0 as | ∪ π| → 0. Hence, by Theorem 6.1 f satisfies condition (2.28), i.e., belongs to AC. This and (9.30) prove the required statement (9.28). Next, let f ∈ AC (⊂ V˙1d ). We prove that L(AC) ⊂ W˙ 1d , where L sends f ∈ V˙1d to its class of equivalence in L ∞ , and that |L( f )|W d ≤ c| f |V d 1

(9.31)

1

for some c depending only on d. This and (9.28) will show that L| AC is an isomorphism of AC onto W˙ 1d . We start with the embeddings AC ⊂ (V˙ pk )0 ⊂ C implying that AC ⊂ N V˙1d , see Section 5 and Corollary 2.2 (c). The latter, in turn, implies that L( f )| Q ∈ BV d (Q), Q ⊂ Q d , and, moreover, | f |V d (Q) ≈ |L( f )| BV d (Q) 1

with constants of equivalence depending only on d, see (9.14), (9.17). Hence, for every π ∈ Π ∑ ∑ |L( f )| BV d (Q) ≤ c(d) | f |V d (Q) . 1

Q∈π

(9.32)

Q∈π

Choosing for every Q ∈ π a packing π Q ∈ Π (Q) such that ∑ ε | f |V d (Q) ≤ E d ( f ; R) + , 1 card π R∈π Q

where ε > 0 is arbitrary small, we obtain from (9.32) that ( ) ∑ ∑ var D α L( f ) ≤ c(d) E d ( f ; R) + ε , |α|=k

∪π

R∈π˜

where π˜ := ∪ Q∈π π Q is a packing. Sending |π | ≥ |π˜ | to zero we conclude from here and (2.28) that for all closed sets S being unions of nonoverlapping cubes, hence, for all Lebesgue measurable sets S lim var D α L( f ) = 0,

|S|→0 S

|α| = d.

By the Radon–Nikodym theorem, see, e.g., [22, Sec. III.10], ∫ (D α L( f ))(S) = f α d x, |α| = d, S

for all Lebesgue measurable sets S ⊂ Q d and some f α ∈ L 1 . In other words, D α L( f ) ∈ L 1 for all |α| = d and |L( f )| BV d = |L f |W d . 1 Hence, L maps f ∈ AC in W˙ 1d and (9.31) is fulfilled. This proves that L| AC : AC → W˙ 1d is an isomorphism.

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55

Finally, presentation (2.30) is valid for continuous representatives f of elements f˚ ∈ W˙ 1d , see, e.g., [36, Sec. 1.10]. As we have proved such f ∈ AC, hence, this completes the proof of the corollary. □ 10. Proof of Theorem 2.24 10.1. Auxiliary result d k k k Let κ : V˙ p∞ → V p∞ := V˙ p∞ /Pk−1 be the quotient map defining the latter space. We then

set ) ( L˜ := κ ◦ L|V˙ k ,

(10.1)

p;S

k where L : V˙ pk → V˙ p∞ is the surjective map of norm one sending a function in V˙ pk to its equivalence class in L ∞ k k By the definition, L˜ : V˙ p;S → V p∞ is a bounded surjective linear map of norm 1. (Recall k k ˙ ˙ that V p;S ⊂ V p is the subspace of functions vanishing on the interpolating set S and that we k naturally identify V˙ p;S with V pk , see Section 4.1.1.) k In the next result, we use the duality between V p;S (≡ V pk ) and U pk established in k k Theorem 2.25 and between V p∞ and U p∞ established in [9, Th. 2.6]. k k Proposition 10.1. The map L˜ : V˙ p;S → V p∞ is weak∗ continuous with respect to the weak∗ k k ˙ topologies on V p;S and V p∞ . k Proof. It suffices to check that L˜ −1 (U ) ⊂ V˙ p;S is weak∗ closed for each set U of the form k k { f ∈ V p∞ : | f (g)| ≤ 1}, where g ∈ U p∞ \ {0}. Since L˜ −1 (U ) is a convex absorbing subset k ˙ of V p;S , by the Krein–Smulian theorem, see, e.g., [22, Thm. V.5.7], it suffices to check that k L˜ −1 (U ) ∩ B(V˙ p;S ) is closed in the weak∗ topology. k Let { f α }α∈Λ be a net of functions in L˜ −1 (U )∩ B(V˙ p;S ) converging in the weak∗ topology to a k function f ∈ B(V˙ p;S ). According to Theorem 2.1, the space V˙ pk consists of functions of the first Baire class (pointwise limits of continuous on Q d functions). Moreover, due to Lemma 4.1, the k closed ball B(V˙ p;S ) is compact in the topology of pointwise convergence on Q d . In particular, the net { f α }α∈Λ converges pointwise to f . Since, in addition, supα ∥ f α ∥∞ ≤ 1, the Rosenthal Main Theorem [41] implies that ∫ ∫ lim f α dµ = f dµ (10.2) α

Qd

Qd

for all signed Borel measures µ on Q d . d Now we apply (10.2) to dµ = h d x with h ∈ Lˆ 1 := {g ∈ L 1 : g ⊥ Pk−1 }. Since every k ˆ function of L 1 is an atom for the space U p∞ , see the text before Theorem 2.24 in Section 2.3, k k k k Lˆ 1 := (U p∞ )0 ⊂ U p∞ . Hence, we can use the duality pairing for (V p∞ , U p∞ ), see [9, Th. 2.6], to get ∫ ∫ ˜ f )](h). ˜ f α )](h) := lim lim[ L( fα h d x = f h d x =: [ L( (10.3) α

α

Qd

Qd

k k k Next, by the definition U p∞ is the completion of (U p∞ )0 ; hence, for every g ∈ U p∞ there k ˆ is a sequence {gn }n∈N ⊂ L 1 ⊂ U p∞ such that ∥g − gn ∥U p∞ → 0 as n → ∞. k

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Applying (10.3) to h = gn we then obtain ( ) ˜ f α )](gn − g) + [ L( ˜ f α )](g) − [ L( ˜ f )](g) = [ L( ˜ f )](gn − g). lim [ L( α ( ) k ˜ f ), L( ˜ f α ) ∈ L˜ B(V˙ k ) ⊆ B(V p∞ Further, since by our assumptions L( ) for all α ∈ Λ, p;S ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ˜ ⏐ ˜ → 0 as n → ∞. f )](gn − g)⏐ ≤ 2∥gn − g∥U p∞ f α )](gn − g)⏐ + ⏐[ L( lim sup ⏐[ L( k α

This and the previous equation imply that ⏐ ⏐ ˜ f α )](g) − [ L( ˜ f )](g)⏐ lim sup ⏐[ L( α ( ⏐ ⏐ ⏐ ⏐) ⏐ ˜ ⏐ ⏐ ˜ ⏐ ≤ lim lim sup ⏐[ L( f α )](gn − g)⏐ + ⏐[ L( f )](gn − g)⏐ = 0, n→∞

α

k ˜ f α )](g) = [ L( ˜ f )](g) for all g ∈ U p∞ i.e., limα [ L( . k k ˜ ˜ f ) also belongs to U ∩ B(V p∞ But L( f α ) ∈ U ∩ B(V p∞ ) for all α ∈ Λ; hence, L( ) since ∗ k this set is closed in the weak topology of V p∞ . In particular, k k ˜ f )) ∩ B(V˙ p;S f ∈ L˜ −1 ( L( ) ⊂ L˜ −1 (U ) ∩ B(V˙ p;S ). k This shows that the set L˜ −1 (U ) ⊂ V˙ p;S is weak∗ closed, as required. The proof of the proposition is complete. □

10.2. Proof of Theorem 2.24 (a) We have to prove that U pk is a Banach space. Since U pk is the completion of the space ((U pk )0 , ∥ · ∥(U pk )0 ), it suffices to prove that ∥ · ∥(U pk )0 is a norm, i.e., that if ∥g∥(U pk )0 = 0 for some g ∈ (U pk )0 , then g = 0. To prove this, first we show that for every f ∈ V˙ pk (⊂ ℓ∞ ) and g ∈ (U pk )0 (⊂ ℓ1 ) ⏐ ⏐ ⏐ ⏐∑ ⏐ ⏐ ⏐ ≤ | f | k ∥g∥ k . ⏐ (10.4) f (x)g(x) Vp Up ⏐ ⏐ ⏐ ⏐x∈Q d d Let m Q ∈ Pk−1 , Q ⊂ Q d , be such that

∥ f − m Q ; Q∥∞ = E k ( f ; Q). ∑ Then for bπ := Q∈π c Q a Q we obtain by the definition of (k, p)-atoms ∑ ∑ ∑ f (x)bπ (x) = cQ ( f (x) − m Q (x)) a Q (x). x∈Q d

Q∈π

x∈Q

Applying the H¨older inequality we derive from here ⏐ ⏐ ⎛ ⎞ 1′ ⎛ ⎞ 1p ⏐ ⏐ p ∑ ∑ ⏐∑ ⏐ ( ) ′ p ⏐ f (x)bπ (x) ⏐⏐ ≤ ⎝ |c Q | p ⎠ ⎝ ∥ f − m Q ; Q∥∞ ∥a Q ∥1 ⎠ ⏐ ⏐ x∈Q d ⏐ Q∈π Q∈π ⎞ 1p

⎛ ≤ [bπ ] p′ ⎝

∑ Q∈π

E k ( f ; Q) p ⎠ ≤ [bπ ] p′ | f |V pk .

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57

Since g ∈ (U pk )0 can be presented as a finite sum of (k, p)-chains bπ , this gives ⏐ ( ⏐ ) ⏐ ⏐∑ ∑ ⏐ ⏐ ⏐ [bπ ] p′ | f |V pk = ∥g∥U pk | f |V pk ; f (x)g(x) ⏐⏐ ≤ inf ⏐ ⏐ ⏐ x∈Q d π here infimum is taken over all such presentations of g. From this inequality and the equality ∥g∥U pk = 0 we deduce that ∑ f (x)g(x) = 0 for all f ∈ V˙ pk .

(10.5)

x∈Q d

In particular, since C ∞ ⊂ V˙ kp ⊂ V˙ pk , see Section 5, Eq. (10.5) is valid for all f ∈ C ∞ . Now assume, on the contrary, that g satisfies (10.5) but g ̸= 0. Let X ⊂ Q d be a finite subset of supp g such that ∑ 1 |g(x)| ≥ ∥g∥1 (> 0). 2 x∈X Clearly, there exists a function ϕ X ⊂ C ∞ such that ϕ X (x) = sgn(g(x)),

x ∈ X,

and

∥ϕ X ∥∞ ≤ 1.

From here and (10.5) we obtain the contradiction ⏐ ⏐ ⏐ ⏐∑ ∑ ∑ 1 1 ⏐ ⏐ |ϕ X (x)||g(x)| < ∥g∥1 . ϕ X (x)g(x)⏐ ≤ ∥g∥1 ≤ |g(x)| = ⏐ ⏐ ⏐ 2 2 d x∈X x∈X x∈Q \X

Hence, g = 0 and the proof of part (a) of the theorem is complete. (b) We have to prove that B(U pk ) is the closure of the symmetric convex hull of the set k {bπ ∈ (U pk )0 : [bπ ] p′ ≤ 1}. This, in fact, is proved in Theorem 2.5 of [9] for B(U p∞ ). The proof can be easily adapted to the required case. k (c) We have to prove that U pk is nonseparable and contains a separable subspace Uˆ pk ≡ U p∞ k ˆk ∗ ∼ such that (U p /U p ) = ℓ p . Let us show that U pk is nonseparable. To this end, as in Lemma 4.1 we take an interpolating d set S ⊂ Q d for the space Pk−1 and a function cx : S → R, x ∈ Q d , such that ∑ d m(x) = cx (s)m(s) for every m ∈ Pk−1 . s∈S

Further, we set ∑ cx (s)δ S , δx′ := δx − s∈S

where δ y is the delta-function at y ∈ Rd . If x ̸= y ∈ Q d \ S, then by definition ∥δx′ − δ ′y ∥U pk :=

1

1

| f (x) − f (y)| ≥ 2− p | f x,y (x) − f x,y (y)| = 2 p′ ;

sup k ;| f | ≤1 f ∈V˙ p;S Vk p

here f x,y (x) = 1, f x,y (y) = −1 and f x,y (z) = 0 otherwise. Hence, U pk contains a discrete uncountable subset, i.e., it is nonseparable.

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Next, let us show that U pk contains a separable subspace Uˆ pk such that k Uˆ pk ≡ U p∞

and

(U pk /Uˆ pk )∗ ∼ = ℓp.

k k In fact, since L˜ : V˙ p;S → V p∞ is weak∗ continuous and surjective, see Proposition 10.1, k ˜ see, e.g., there is a bounded linear injective map T : U p∞ → U pk such that T ∗ = L, [22, VI.9.13]. We set k Uˆ pk := T (U p∞ ) (10.6) k and prove that T : U p∞ → (Uˆ pk , ∥ · ∥U pk ) is isometry. k k Due to Theorem 2.5 (c) L˜ maps the Banach space N V˙ pk ∩ V˙ p;S isometrically onto V p∞ . k k Further, let u ∈ U p∞ \ {0} and f u ∈ V p∞ be such that f u (u) = ∥u∥U p∞ and ∥ f ∥V p∞ = 1. k k k ˜ f˜u ) = f u and | f˜u | k = 1. In particular, we Then there exists f˜u ∈ N V˙ pk ∩ V˙ p;S such that L( Vp have ˜ f˜u )(u)| = | f u (u)| = ∥u∥ k . ∥T (u)∥ k ≥ | f˜u (T u)| = | L( Up

U p∞

On the other hand, ˜ · ∥u∥ k = ∥u∥ k ∥T (u)∥U pk ≤ ∥T ∥ · ∥u∥U p∞ = ∥ L∥ k U p∞ U p∞ and therefore ∥T (u)∥U pk = ∥u∥U p∞ k

for all

k u ∈ U p∞ ,

k i.e., T : U p∞ → (Uˆ pk , ∥ · ∥U pk ) is an isometric isomorphism and Uˆ pk is a closed subspace of U pk . k It is separable because U p∞ is by [9, Th. 2.5(b)]. Further, by the definition of dual to a factor-space, see, e.g., [22, II.4.18], (U pk /Uˆ pk )∗ is k isomorphic to the annihilator (Uˆ pk )⊥ ⊂ V pk of Uˆ pk . Moreover, since T : U p∞ → Uˆ pk is an k ⊥ k isomorphism, an element f ∈ (Uˆ p ) iff for every u ∈ U p∞

˜ f ))(u) = 0, f (T (u)) = ( L( ˜ ∼ i.e., iff f ∈ ker( L) = ℓ p (see Theorem 2.5(b)). This shows that (U pk /Uˆ pk )∗ ∼ = ℓ p and completes the proof of the theorem. 11. Proof of Theorem 2.25 In the proof we use the following auxiliary result. Let f ∈ ℓ∞ (Q). There is a sequence of functions {gn }∈N ∈ ℓ1 (Q) such that ∑ E k ( f ; Q) = lim f (x)gn (x); (11.1) n→∞ x∈Q ∑ ∥gn ∥ℓ1 (Q) = 1 and x α gn (x) = 0, |α| ≤ k − 1, for all n ∈ N. (11.2)

Lemma 11.1.

x∈Q d Proof. Let ℓˆ1 (Q) := {g ∈ ℓ1 (Q) : g ⊥ Pk−1 }. By the Hahn–Banach theorem ℓˆ1 (Q)∗ ≡ d ℓ∞ (Q)/Pk−1 . Since the best approximation in (11.1) is the norm of the image of f in d ℓ∞ (Q)/Pk−1 | Q , the definition of the norm of a bounded linear functional on ℓˆ1 (Q) implies existence of a sequence {gn }n∈N ⊂ ℓˆ1 (Q) of elements of norm 1 satisfying (11.1) □

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59

Proof of Theorem 2.25. By definition, (U pk )0 ⊂ ℓˆ1 . Further, every function f ∈ ℓˆ1 is a (k, p)-chain subordinate to the packing π = {Q d }. d In fact, a function f = c Q d a Q d , where c Q d := ∥ f ∥1 and a Q d := f /∥ f ∥1 , vanishes on Pk−1 k and, moreover, ∥a Q d ∥1 = 1. Hence, by the definition of the seminorm of U p , ∥ f ∥U pk ≤ |c Q d | = ∥ f ∥1 .

(11.3)

In other words, the linear embedding E : ℓˆ1 ↪→ U pk

(11.4)

holds with the embedding constant 1 and has dense image. Passing to the adjoint map we obtain that d E ∗ : (U pk )∗ ↪→ ℓˆ∗1 ≡ ℓ∞ /Pk−1 d is a linear injection of norm ≤ 1. On the other hand, V pk is contained in ℓ∞ /Pk−1 . ∗ k ∗ k ∗ Further, we show that range(E ) is in V p and that the linear map E : (U p ) → V pk is of norm ≤ 1. d To this end, for ℓ ∈ (U pk )∗ we denote by f ℓ ∈ ℓ∞ an element whose image in ℓ∞ /Pk−1 1 ∗ d coincides with E (ℓ). Then we take for every Q ⊂ Q and ε ∈ (0, 2 ) a (k, p)-atom denoted by a˜ Q such that ∑ (1 − ε)E k ( f ℓ ; Q) ≤ f ℓ (x)a˜ Q (x) ≤ E k ( f ℓ ; Q); (11.5) x∈Q

its existence directly follows from Lemma 11.1 ∑and the definition of (k, p)-atoms. Then for a (k, p)-chain b˜π given by b˜π := Q∈π c Q a˜ Q we get from (11.5) ∑ ∑ [E ∗ (ℓ)](b˜π ) = b˜π (x) f ℓ (x) ≥ (1 − ε) c Q E k ( f ℓ ; Q). x∈Q d

Q∈π

This, in turn, implies ∑ c Q E k ( f ℓ ; Q) ≤ (1 − ε)−1 ∥E ∗ (ℓ)∥1 ∥b˜π ∥U pk ≤ (1 − ε)−1 ∥ℓ∥(U pk )∗ ∥b˜π ∥U pk Q∈π

⎞ 1′

⎛ ≤ (1 − ε)−1 ⎝

p

∑

′

|c Q | p ⎠

∥ℓ∥(U pk )∗ .

Q∈π

Sending here ε → 0 and then taking supremum over all sequences (c Q ) Q∈π of the ℓ p′ (π ) norm 1 and supremum over all π we conclude that ⎛ ⎞ 1p ∑ | f ℓ |V pk := sup ⎝ E k ( f ℓ ; Q) p ⎠ ≤ ∥ℓ∥(U pk )∗ . π

Q∈π

Hence, E (ℓ) ∈ for every ℓ ∈ (U pk )∗ and E ∗ : (U pk )∗ ↪→ V pk is a linear injection of norm ≤ 1. Next, let us show that there is a linear injection F of norm ≤ 1 ∗

V pk

F : V pk ↪→ (U pk )∗

(11.6)

such that F E ∗ = id|(U pk )∗ .

(11.7)

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Actually, let f ∈ V˙ pk and ℓ f : (U pk )0 → R be a linear functional given for g ∈ (U pk )0 by ∑ ℓ f (g) := f (x)g(x). (11.8) x∈Q d

Due to (10.4) |ℓ f (g)| ≤ | f |V pk ∥g∥U pk . Thus, ℓ f continuously extends to a linear functional from (U pk )∗ (denoted by the same symbol) d and the linear map F : V pk → (U pk )∗ , { f } + Pk−1 ↦→ ℓ f , is of norm ≤ 1. Further, we show that F is an injection. Indeed, let ℓ f = 0 for some f ∈ V˙ pk . Since (U pk )0 = ℓˆ1 and V˙ pk ⊂ ℓ∞ , equality (11.8) implies that ℓ f | k 0 determines the trivial functional on (ℓˆ1 )∗ ≡ ℓ∞ /P d . Hence, f ∈ P d , k−1

(U p )

k−1

i.e., f determines the zero element of the factor-space V pk , as required. Further, by definitions of E ∗ and F we have for each h ∈ (U pk )∗ and g ∈ (U pk )0 ∑ [F E ∗ (h)](g) = ℓ E ∗ (h) (g) = E ∗ (h)(x)g(x) = h(E(g)) = h(g). x∈Q d

Proofs of (11.6) and (11.7) are complete. In turn, the established results mean that range(E ∗ ) = V pk , range(F) = (U pk )∗ and F and E ∗ are isometries. The theorem is proved. □ 12. Proof of Theorem 2.26 (a) We have to prove that the map sending a chain b ∈ (U pk )0 (⊂ ℓ1 ) to a discrete measure ˜pk )0 (⊂ M), where µb ({x}) := b(x), x ∈ Q d , extends to an isometric embedding I of µb ∈ (U ˜pk . U pk into U ˜pk )0 , see Sections 2.3, 2.4, we obtain In fact, by definitions of seminorms of (U pk )0 and (U ∥µb ∥U˜pk ≤ ∥b∥U pk

for all

b ∈ (U pk )0 ,

(12.1)

˜pk )0 extends by continuity to a bounded linear map i.e., the map (U pk )0 ∋ b ↦→ µb ∈ (U ˜pk of norm ≤ 1. I : U pk → U The fact that I is an isometry is a consequence of the following properties of the introduced ˜pk → U pk that will be established in part (b) of the proof: below operator E : U E ◦ I = id

and

∥E∥ ≤ 1.

Actually, these imply ∥u∥U pk = ∥E(I(u))∥U pk ≤ ∥I(u)∥U˜pk ≤ ∥u∥U pk

for all

u ∈ U pk .

˜pk is an isometric embedding. Hence, I : U pk → U ˜pk → U pk such that (b) We have to prove that there exists a linear continuous surjection E : U ˜pk )0 = {0} ker(E) ∩ (U

and

E ◦ I = id.

˜pk )0 by To define such E we use a bilinear form given for v ∈ V˙ pk and µ ∈ (U ∫ ⟨v, µ⟩ := v dµ. Qd

(12.2)

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61

Since due to Theorem 2.1 V˙ pk consists of bounded functions of the first Baire class, this form is correctly defined. As we show now it can be continuously extended to the seminormed Banach ˜pk . (We will retain notation ⟨·, ·⟩ for the extension.) space V˙ pk × U To this end it suffices to establish the inequality ˜pk )0 , |⟨v, µ⟩| ≤ |v| k ∥µ∥ ˜k , (v, µ) ∈ V˙ pk × (U (12.3) Vp

Up

whose proof repeats line by line the proof of inequality (10.4) with sum replaced by integral, a Q ∈ ℓ1 by µ Q ∈ M and ∥ · ∥ by ∥ · ∥ M . We leave the details to the readers. ˜pk and, hence, for every such µ the map Thus, inequality (12.3) is valid for all µ ∈ U ℓµ : v ↦→ ⟨v, µ⟩,

k v ∈ V˙ p;S (≡ V pk ),

determines a linear continuous functional from (V pk )∗ . Now we define the required map E by the formula ˜pk . E(µ) := ℓµ , µ ∈ U ˜pk U

(12.4)

(V pk )∗

Then E acts linearly from to and is of norm ≤ 1. We have to prove that the range of E coincides with U pk (⊂ (U pk )∗∗ ≡ (V pk )∗ ). To this end we establish the following: ˜pk )0 is weak∗ continuous on V pk with respect Each bounded linear functional ℓµ with µ ∈ (U ∗ k ∗ to the weak topology induced by the duality (U p ) ≡ V pk . k ∗ It suffices to check that ℓ−1 µ ([−1, 1]) ⊂ V p , ℓµ := E(µ), is weak closed. In fact, since k ℓ−1 µ ([−1, 1]) is a convex absorbing subset of V p , it suffices by the Krein–Smulian theorem, −1 see, e.g., [22, Thm. V.5.7], to check that ℓµ ([−1, 1]) ∩ B(V pk ) is closed in the weak∗ topology. k In the proof of this, without loss of generality we identify V pk with V˙ p;S ⊂ V˙ pk , see Section 4.1.1, k k so that B(V p ) = B(V˙ p;S ). ∗ ˙k Let { f α }α∈Λ be a net of functions in ℓ−1 µ ([−1, 1]) ∩ B( V p;S ) converging in the weak k ). Since V˙ pk consists of functions of the first Baire class, see topology to a function f ∈ B(V˙ p;S k Theorem 2.1, and the closed ball B(V˙ p;S ) is compact in the topology of pointwise convergence d on Q (see Lemma 4.1), the net { f α }α∈Λ converges pointwise to f ; in addition, supα ∥ f α ∥∞ ≤ 1 and ∥ℓµ ∥ ≤ 1. These and the Rosenthal Main Theorem [41] imply that ∫ ∫ ∫ ℓµ ( f ) := f dµ = lim f α dµ = lim f α dµ = lim ℓµ ( f α ) (∈ [−1, 1]); (12.5) Qd

Qd

α

α

Qd

α

and ℓµ ∈ is weak continuous. hence, f ∈ Since the subspace of weak∗ continuous functionals in (V pk )∗ coincides with U pk , we conclude ˜pk )0 . Moreover, U pk is a closed subspace of (V pk )∗ and E is that E(µ) ∈ U pk for all µ ∈ (U k 0 ˜ ˜pk . These facts imply that range(E) ⊂ U pk . continuous on (U p ) that is dense in U Next, we show that ˜pk )0 = {0}. ker(E) ∩ (U (12.6) ℓ−1 µ ([−1, 1])

(V pk )∗

∗

˜pk )0 . Then for each f ∈ C ∞ ⊂ V˙ pk In fact, let µ ∈ ker(E) ∩ (U ∫ ( ) E(µ) ( f ) := f dµ = 0. Qd

Since C ∫

∞

is dense in C in the topology of uniform convergence, the latter implies that f dµ = 0

Qd

for all

f ∈ C.

(12.7)

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According to the Riesz representation theorem C ∗ ≡ (M, ∥ · ∥ M ) with the duality defined by integration with respect to the corresponding measure; hence, (12.7) implies that µ = 0 proving (12.6). Finally, we prove that E ◦ I = id.

(12.8)

In fact, we have for b ∈ (⊂ ℓ1 ) and all v ∈ (⊂ ℓ∞ ) ∫ ∑ ( ) E(I(b)) (v) = v dµb = v(x)b(x) = b(v). k V˙ p;S

(U pk )0

Q

x∈Q d

Since by Theorem 2.25 (U pk )∗ ≡ V pk , the latter implies that E(I(b)) = b

for all

b ∈ (U pk )0 .

From here using that E and I are continuous maps and that (U pk )0 is dense in U pk we obtain the required identity (12.8). In particular, this implies that range(E) = U pk , i.e., surjectivity of E, and completes the proof of part (b) and hence of the theorem. 13. Proofs of Theorem 2.27 and Corollary 2.28 Proof of Theorem 2.27. We have to prove that if d s := < k, 1 < p < ∞, p then ∼ Uk. ( V k )∗ = p

p

Since the relation ( V k )∗ ∼ = Uk p∞

p∞

(13.1)

(13.2)

(13.3)

under condition (13.1) has been just proved in [9, Th. 2.7] and the spaces under consideration are defined similarly to those in (13.3), in the forthcoming proof we can use some basic arguments of the cited paper with some trivial modification of notations. In the following text, as before, we identify U pk with its image under the natural embedding k U p ↪→ (U pk )∗∗ and V pk with (U pk )∗ , see Theorem 2.25; hence, we regard U pk as a linear subspace of (V pk )∗ (= ((U pk )∗ )∗ ). Further, i : Vkp ↪→ V pk is the natural embedding and i∗ : (V pk )∗ → (Vkp )∗ is its adjoint. Proposition 13.1. (a) i∗ is a surjective linear map of norm one which maps U pk isomorphically onto a closed subspace of (Vkp )∗ . (b) The image i∗ (B(U pk )) is a dense subset of B((Vkp )∗ ) in the weak∗ topology of (Vkp )∗ . Proof. The proof that i∗ : (V pk )∗ → (Vkp )∗ is a surjective linear map of norm one repeats line by line the proof of the similar assertion of Proposition 6.1 in [9]. The same is true for part (b) of Proposition 13.1. To prove the remaining statement of part (a) of the proposition we use the following analog of Lemma 6.2 of [9].

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Lemma 13.2.

63

The subspace Vkp is weak∗ dense in the space V pk (= (U pk )∗ ).

Proof. The result is a straightforward corollary of Theorem 2.3.

□

Now, to prove that i∗ maps U pk isomorphically onto a closed subspace of (Vkp )∗ , we have to show that there is a constant c ∈ (0, 1) such that ∥i∗ (v)∥(Vkp )∗ ≥ c∥v∥U pk for all v ∈ U pk . In fact, let u ∈ U pk \ {0}. By the Hahn–Banach theorem there exists f ∈ V pk such that ∥ f ∥V pk = 1 and f (u) = ∥u∥U pk . By Theorem 2.3 there exists a sequence { f n }n∈N ⊂ Vkp weak∗ converging to f such that sup ∥ f n ∥V pk ≤ C∥ f ∥V pk = C n

for some C ≥ 1 independent of f . These imply that ∥v∥U pk = | f (v)| = lim | f n (v)| ≤ C sup |g(v)| n→∞

g∈B(Vkp )

= C sup |(i (v))(g)| := C∥i∗ (v)∥(Vkp )∗ , ∗

g∈B(Vkp )

as required. This completes the proof of Proposition 13.1. □ Our next result is an analog of Proposition 6.3 of [9]. In its formulation, we use ˜pk )0 vanishing on P d with a Theorem 2.26 to identify the vector space of measures Mˆ := (U k−1 k k k subspace of U p . Let B¯ p stand for the closure in U p of the set of (k, p)-chains ˆ ˜pk )0 : [bπ ] p′ ≤ 1} ⊂ M, B kp := {bπ ∈ (U

(13.4)

see (2.31) for the definition. Proposition 13.3.

i∗ (B¯ kp ) is a subset of B((Vkp )∗ ) compact in the weak∗ topology of (Vkp )∗ .

Proof. As in the proof of [9, Prop. 6.3], the required result follows from the next one. Statement 13.4. If {bn }i∈N ⊂ B kp is such that the sequence {i∗ (bn )}n∈N weak∗ converges in B((Vkp )∗ ), then its limit belongs to i∗ (B¯ kp ). Proof. Let bn have the form bn :=

N (n) ∑

cin µ Q in ,

n ∈ N,

i=1

where πn := {Q in : 1 ≤ i ≤ N (n)} ∈ Π is a packing. As in the cited proposition we assume without loss of generality that n |Q i+1 | ≤ |Q in |,

1 ≤ i < N (n).

Further, we extend sequences πn , {cin } and {µ Q in } by setting Q in := {0},

cin := 0,

µ Q in := 0

for i > N (n).

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Hence, we write bn :=

∞ ∑

cin µ Q in ,

n ∈ N.

i=1

Lemma 13.5. There is an infinite subsequence {bn }n∈J , J ⊂ N, such that for every i ∈ N the following is true. (a) {Q in }n∈J converges in the Hausdorff metric to a closed subcube of Q d denoted by Q i ; (b) {i∗ (µ Q in )}n∈N ⊂ B((Vkp )∗ ) converges in the weak∗ topology of B((Vkp )∗ ); (c) if the limiting cube Q i has a nonempty interior, then the sequence of measures {µ Q in }n∈J converges in the weak∗ topology of M (regarded as the dual space of C := C(Q d )); (d) the sequence {cn := (cin )i∈N }n∈J of vectors from B(ℓ p′ (N)) converges in the weak∗ topology of ℓ p′ (N) (= ℓ p (N)∗ ) to a vector denoted by c (∈ B(ℓ p′ (N))). Proof. The proof of the result repeats line by line the proof of [9, Lm. 6.5]. □ Hence, without loss of generality we can assume that the sequence {bn } ⊂ B kp of (k, p)-chains satisfies the assertions of Lemma 13.5. In particular, there are closed cubes Q i ⊂ Q d , i ∈ N, such that in the Hausdorff metric Q i = lim Q in .

(13.5)

n→∞

Since for each n ∈ N the cubes Q in , i ∈ N, are nonoverlapping and their volumes form a nonincreasing sequence, the same is true for the family of cubes {Q i }i∈N . Thus, for every i ∈N Q˚ i ∩ Q˚ i+1 = ∅ and

|Q i | ≥ |Q i+1 |.

(13.6)

Now we let N = ∞ if |Q i | ̸= 0 for all i ∈ N, otherwise, N be the minimal element of the set of integers n ∈ Z+ such that |Q i | = 0

for i > n.

(13.7)

Then due to Lemma 13.5 (c) for N ̸= 0 there are measures µi ∈ M, 1 ≤ i < N + 1, such that in the weak∗ topology of M µi = lim µ Q in . n→∞

(13.8)

Lemma 13.6. (1) If N ̸= ∞ and i > N , then in the weak∗ topology of B((Vkp )∗ ) lim i ∗ (µ Q in ) = 0.

n→∞

(13.9)

(2) If N ̸= 0 and 1 ≤ i < N + 1, then the measure µi is a (k, p)-atom subordinate to Q i . Proof. The proof of part (1) repeats with the corresponding change of notations and definitions the analogous assertion in Lemma 6.6 of [9]. (2) First, due to (13.8) ∥µi ∥ M ≤ 1 as atoms µ Q in belong to the closed unit ball of M = C ∗ which is weak∗ compact.

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65

Next, we prove that supp µi ⊂ Q i

d µi ⊥ Pk−1 .

and

(13.10)

If, on the contrary, supp µi \ Q i ̸= ∅, then there is a continuous function f ∈ C such that (supp f ) ∩ Q i = ∅ and ∫ f dµi ̸= 0. (13.11) Qd

However, Q in → Q i in the Hausdorff metric as n → ∞ and therefore Q in ∩ supp f = ∅ for all sufficiently large n. This and condition (c) of Lemma 13.5 imply that ∫ ∫ 0 = lim f dµ Q in = f dµi n→∞

Qd

Q

in contradiction with (13.11). d To prove the second assertion of (13.10) we note that µ Q in ⊥ Pk−1 for all i and n. Then d according to condition (c) of Lemma 13.5 we have for each m ∈ Pk−1 ∫ ∫ mdµi = mdµ Q in = 0, Qd

Qd

as required. Hence, µi (in the sequel denoted by µ Q i ) is a (k, p)-atom subordinate to Q i . The proof of the lemma is complete. □ Now we show that for 1 ≤ N < ∞ v N :=

N ∑

ci µ Q i ∈ B kp .

(13.12)

i=1

In fact, by Lemma 13.6 (2) and (13.6) µ Q i are (k, p)-atoms and {Q i }1≤i≤N is a packing. Moreover, by Lemma 13.5 (d) the (k, p)-atom v N satisfies [v N ] p′ = ∥c∥ p′ ≤ 1, as required. Further, for N = ∞ v∞ :=

∞ ∑

ci µ Q i ∈ B¯ kp

(13.13)

i=1

In fact, by Lemma 13.5 (d) and 13.6 (2), and inequality (12.3),  m  ( m ) 1′ ∑  p ∑ ′   →0 ci µ Q i  ≤ |ci | p    k i=ℓ

Up

i=ℓ

as ℓ, m → ∞, i.e., the series in (13.13) converges in U pk . Moreover, its partial sums belong to B kp , cf. (13.12), hence, v∞ belongs to the closure of B kp . In the remaining case of N = 0 we set v0 := 0.

(13.14)

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Now the proof of Statement 13.4 will be completed if we show that lim i∗ (bn ) = i∗ (v N )

(13.15)

n→∞

in the weak∗ topology of B((Vkp )∗ ). However, the derivation of (13.15) repeats line by line the proof of a similar statement in [9], see equation (6.23) there. The proof of Statement 13.4 is complete. □ As it was mentioned above this result implies Proposition 13.3. □ Now, we complete the proof of Theorem 2.27 beginning with the following: Lemma 13.7. The space Vkp embeds isometrically in the space C(i∗ (B¯ kp )) of continuous functions on the (metrizable) compact space i∗ (B¯ kp ). Proof. By Theorem 2.24 (b) B(U pk ) is the closure of the symmetric convex hull of the set {bπ ∈ (U pk )0 : [bπ ] p′ ≤ 1}. Since the latter set is a subset of B kp ⊂ B(U pk ), see (13.4), B(U pk ) is the closure of the symmetric convex hull of B kp , denoted by sc(B kp ). As in [9, Lm. 6.7] this implies that the set i∗ (sc(B¯ kp )) = sc(i∗ (B¯ kp ))

(13.16)

is weak dense in see Proposition 13.3, and for every element v ∈ bounded linear functional on (Vkp )∗ B((Vkp )∗ ),

∗

∥v∥Vkp =

sup

|v(v ∗ )|.

V kp

regarded as a (13.17)

v ∗ ∈i∗ (B¯ kp )

Since v|i∗ (B¯ kp ) is a continuous function on i∗ (B¯ kp ) in the weak∗ topology induced from B((Vkp )∗ ) and its supremum norm equals ∥v∥Vkp , the map k

Vp

∋ v ↦→ v(v ∗ ),

v ∗ ∈ i∗ (B¯ kp ),

is a linear isometric embedding of Vkp in C(i∗ (B¯ kp )).

□

Now let v ∗ be a linear continuous functional on the space Vkp regarded as the closed subspace of the space C(i∗ (B¯ kp )). By the Hahn–Banach theorem v ∗ can be extended to a linear continuous functional, say, vˆ ∗ on this space with the same norm. In turn, by the Riesz representation theorem there is a regular finite (signed) Borel measure on the compact space i∗ (B¯ kp ) denoted by µv∗ that represents vˆ ∗ . This implies that ∫ v(v ∗ ) = v dµv∗ , v ∈ Vkp . (13.18) i∗ (B¯ kp )

Now we use this measure to find a similar representation for elements of V pk . Since Vkp is a weak∗ dense subspace of the space V pk , see Lemma 13.2 and Theorem 2.3, for every v ∈ V pk there is a bounded in the Vkp norm sequence {v j } j∈N ⊂ Vkp such that lim v j (u) = v(u),

j→∞

u ∈ U pk .

(13.19)

A. Brudnyi and Y. Brudnyi / Journal of Approximation Theory 251 (2020) 105346

67

Now let τ : i∗ (U pk ) → U pk be the inverse to the injection i∗ |U pk : U pk → (Vkp )∗ , see Proposition 13.1 (a). Making the change of variable u → τ (v ∗ ) we derive from (13.19) lim v j (v ∗ ) = (v ◦ τ )(v ∗ ), v ∗ ∈ i∗ (B¯ k ). (13.20) p

j→∞

Since linear functionals v j : Vkp → R are continuous in the weak∗ topology defined by (Vkp )∗ , their traces to i∗ (B¯ kp ) are continuous functions in the weak∗ topology induced from B((Vkp )∗ ). This implies the following: Lemma 13.8. The function (v ◦ τ )|i∗ (B¯ kp ) is µv∗ -integrable and bounded. Moreover, a function φv∗ : V pk → R given by ∫ φv∗ (v) := v ◦ τ dµv∗

(13.21)

i∗ (B¯ kp )

belongs to (V pk )∗ . Proof. The result is, in fact, proved in Lemma 6.8 of [9].

□

At the next stage we establish weak continuity of φv∗ on V pk regarded as the dual space of see Theorem 2.25. k ∗ To this end it suffices to show that φv−1 ∗ (R) ⊂ V p is weak closed for every closed interval R ⊂ R. Since this preimage is convex, we can use the Krein–Smulian weak∗ closedness ∗ criterion, see, e.g., [22, Thm. V.5.7]. In our case, it asserts that φv−1 ∗ (R) is weak closed iff −1 k Br (0) ∩ φv∗ (R) is for every r > 0; here Br (0) := {v ∈ V p : ∥v∥V pk ≤ r }. k k Without loss of generality we identify (V pk , ∥ · ∥V pk ) with (V˙ p;S , | · |V pk ), where V˙ p;S ⊂ V˙ pk is d d the subspace of functions vanishing on an interpolating set S ⊂ Q for Pk−1 , see Section 4.1.1. Further, let B1 (Q d ) stand for the space of functions on Q d of the first Baire class equipped d ˙k with the topology of pointwise convergence. By Lemma 4.1 Br (0) ∩ φv−1 ∗ (R) ⊂ V p;S ⊂ B1 (Q ) is relatively compact in the latter space and by Theorem F3 of [4] is sequentially dense in its closure. Hence, if v belongs to this closure, then there is a sequence {v j } j∈N ⊂ Br (0) ∩ φv−1 ∗ (R) converging pointwise to v on Q d . Therefore {v j } j∈N satisfies the assumptions of the Rosenthal Main theorem [41] implying that ∫ ∫ v j dµ = v dµ (13.22) lim ∗

U pk ,

j→∞

Qd

Qd

for all finite signed Borel measures on Q d . In particular, this is true for discrete measures defined by functions in ℓˆ1 = (U pk )0 . But (U pk )0 k is dense in U pk and by our definition the sequence {v j } j∈N is bounded in V˙ p;S . Hence, (13.22) implies that lim v j (u) = v(u)

j→∞

for all

u ∈ U pk ;

(13.23)

here we regard v j and v as bounded linear functionals on U pk , see Theorem 2.25. k This means that {v j } j∈N ⊂ V˙ p;S weak∗ converges to v. ∗ To show weak closedness of Br (0) ∩ φv−1 ∗ (R) it remains to prove the following: ∗ k Lemma 13.9. If a sequence {v j } j∈N ⊂ Br (0) ∩ φv−1 ∗ (R) weak converges to some v ∈ V p , −1 then v ∈ Br (0) ∩ φv∗ (R).

68

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Proof. Weak∗ convergence of {v j } j∈N to v implies pointwise convergence of the sequence of functions {v j ◦ τ |i∗ (B¯ kp ) } j∈N to the function v ◦ τ |i∗ (B¯ kp ) , see (13.19). Further, functions of this sequence are µv∗ -measurable and bounded by sup j ∥v j ∥Vκ , see Lemma 13.8. Moreover, by the assumption of Lemma 13.9 sup ∥v j ∥V pk ≤ r

and

φv∗ (v j ) ∈ R, j ∈ N.

(13.24)

j

Therefore, the Lebesgue dominated convergence theorem implies ( ) ∫ ∫ lim φv∗ (v j ) = lim v j ◦ τ dµv∗ = v ◦ τ dµv∗ = φv∗ (v). j→∞

i∗ (B¯ kp )

j→∞

i∗ (B¯ kp )

Since R ⊂ R is closed, the limit on the left-hand side belongs to R, hence, the limit point v ∈ Br (0) ∩ φv−1 □ ∗ (R) as required. Thus, φv∗ is a weak∗ continuous linear functional from (V pk )∗ . By the definition of the weak∗ topology on V pk = (U pk )∗ every weak∗ continuous functional is uniquely determined by an element of U pk , see, e.g., [18, V.1.3], i.e., for some u v∗ ∈ U pk φv∗ (v) = v(i∗ (u v∗ )),

v ∈ V pk .

On the other hand, see (13.21), for all v ∈ Vkp , φv∗ (v) = v(v ∗ ). Moreover, since i∗ |U pk : U pk → (Vkp )∗ is an isomorphic embedding, see Proposition 13.1 (a), Vkp separates points of U pk . These two equalities imply that v ∗ = i∗ (u v∗ ). Thus, every point v ∗ ∈ (Vkp )∗ is the image under i∗ of some point of U pk , i.e., i∗ : U pk → (Vkp )∗ is a surjection. Moreover, i∗ |U pk is also an embedding. Hence, i∗ is an isomorphism of the Banach spaces U pk and (Vkp )∗ . This completes the proof of Theorem 2.27. □ Proof of Corollary 2.28. The result is the combination of Theorems 2.25 and 2.27. □ Acknowledgments We thank the anonymous referees for useful comments improving the presentation of the paper. References [1] C.R. Adams, J.A. Clarkson, On definitions of bounded variation for functions of two variables, Trans. Amer. Math. Soc. 34 (1933) 824–854. [2] C.R. Adams, J.A. Clarkson, Properties of functions f (x, y) of bounded variation, Trans. Amer. Math. Soc. 36 (1934) 711–730. [3] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variations and Discontinuity Problems, Oxford Sci. Publ, 2000. [4] J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978) 845–886.

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