Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

Nonlinear Analysis 192 (2020) 111684

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Maximal commutators and commutators of potential operators in new vanishing Morrey spaces Alexandre Almeida Center for R&D in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

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Article history: Received 5 July 2019 Accepted 28 October 2019 Communicated by Vicentiu D Radulescu MSC: 46E30 42B35 42B25 47B47

abstract We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Morrey spaces Vanishing properties Commutators Maximal functions Fractional integral operators

1. Introduction Morrey spaces Lp,λ were introduced by C. Morrey [27] in connection with the study of local behavior of solutions to partial differential equations. Since then they are widely used in the regularity theory of PDE, including heat equations and Navier–Stokes equations. We refer to [21,23,40,42,43] for further details and references on the role played by Morrey spaces in applications. The theory of Morrey spaces was further developed by Campanato [7] and Peetre [28]. The main properties of these spaces and historical remarks can be found in the books [2,15,30,41,42] and in the overview [31]. We also refer to [3,34,43] for a discussion on Harmonic Analysis in Morrey spaces. For 1 ≤ p < ∞, 0 ≤ λ ≤ n, the classical Morrey space Lp,λ (Rn ) consists of all locally p-integrable functions f on Rn with finite norm ∥f ∥p,λ :=

E-mail address: [email protected]. https://doi.org/10.1016/j.na.2019.111684 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

sup x∈Rn , r>0

λ

r− p ∥f ∥Lp (B(x,r)) .

(1.1)

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One of the intrinsic difficulties in dealing with Morrey spaces Lp,λ (Rn ) is the non-separability of such spaces when λ > 0. The approximation problem by nice functions in this setting has led to the introduction of appropriate subspaces (cf. [6,44,45,47]). In this paper we consider the following subspaces of Lp,λ (Rn ). The class V0 Lp,λ (Rn ) consists of all those functions f ∈ Lp,λ (Rn ) such that lim sup Mp,λ (f ; x, r) = 0,

(V0 )

r→0 x∈Rn

where Mp,λ (f ; x, r) :=

1 rλ

∫

p

|f (y)| dy ,

x ∈ Rn ,

r > 0.

(1.2)

B(x,r)

Similarly, V∞ Lp,λ (Rn ) is the set of all f ∈ Lp,λ (Rn ) such that lim sup Mp,λ (f ; x, r) = 0.

r→∞ x∈Rn

(V∞ )

We also consider the set V (∗) Lp,λ (Rn ) consisting of all functions f ∈ Lp,λ (Rn ) having the vanishing property ∫ p lim AN,p (f ) := lim sup |f (y)| χN (y) dy = 0, (V ∗ ) N →∞

N →∞ x∈Rn

B(x,1)

where χN := χRn \B(0,N ) , N ∈ N. The three vanishing classes defined above are closed sets in Lp,λ (Rn ) with respect to the norm (1.1). The space V0 Lp,λ (Rn ), often called in the literature just by vanishing Morrey space, was already introduced in [9,44,45] motivated by regularity results of elliptic equations. The subspaces V∞ Lp,λ (Rn ) and V (∗) Lp,λ (Rn ) were recently introduced in [6] to study the delicate problem on the approximation of Morrey functions by nice functions. Note that V∞ Lp,λ (Rn ) was independently considered in [46] in the study of interpolation problems. (∗) The subspace V0,∞ Lp,λ (Rn ), collecting those Morrey functions having all the vanishing properties (V0 ), (V∞ ) and (V ∗ ), provides an explicit description of the closure of C0∞ (Rn ) in Morrey norm, see [6, Theorem 5.3 and Corollary 5.4]. There are many papers in the literature on the behavior of classical operators from harmonic analysis in classical Morrey spaces. Few of them are already devoted to the boundedness of operators in V0 Lp,λ (Rn ), including the case of generalized parameters, see [12,18,25,26,29,32,36,37]. Up to author’s knowledge, the study of classical operators in the vanishing Morrey spaces V∞ Lp,λ (Rn ) and V (∗) Lp,λ (Rn ) was only carried out in the recent paper [5] (see also [6, Theorem 3.8] for first boundedness results for convolution operators with integrable kernels on such spaces). Commutator estimates are known to play an important role in many applications in harmonic analysis and partial differential equations, see [10,16,24,33]. In this paper we bring together commutators and vanishing Morrey spaces and show that the vanishing properties (V∞ ) and (V ∗ ) are preserved under the action of maximal commutators and also of commutators of fractional integral operators with BM O coefficients. Notice that norm inequalities for such operators on the whole Morrey space Lp,λ (Rn ) are known, as well as their boundedness in the subspace V0 Lp,λ (Rn ), cf. [12,18,32]. The invariance of the space V (∗) Lp,λ (Rn ) with respect to the maximal commutator is particularly hard to obtain, cf. Theorem 3.5. Section 2 provides some preliminaries on notation, definitions and basic facts on BM O functions and the commutators under consideration. In Section 3 we obtain boundedness results for maximal commutators on the subspaces V∞ Lp,λ (Rn ) (Theorem 3.3) and V (∗) Lp,λ (Rn ) (Theorem 3.5). In Section 4 we derive Adams type results for the (Lp,λ → Lq,λ )- boundedness of fractional maximal commutators (Theorem 4.1) and commutators of Riesz potential operators (Theorem 4.4) on the same subspaces. The very short last section contains some consequences of the previous results. In particular, it emphasizes the invariance of the closure of C0∞ (Rn ) in Morrey norm with respect to those operators.

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2. Preliminaries We use the following notation: B(x, r) is the open ball in Rn centered at x ∈ Rn and radius r > 0. The (Lebesgue) measure of a measurable set E ⊆ Rn is denoted by |E| and χE denotes its characteristic function. We use c as a generic positive constant, i.e., a constant whose value may change with each appearance. The expression A ≲ A means that A ≤ c B for some independent constant c > 0, and A ≈ B means A ≲ B ≲ A. As usual C0∞ (Rn ) stands for the class of all complex-valued infinitely differentiable functions on Rn with compact support, and Lp (Rn ) denotes the classical Lebesgue space equipped with the usual norm. 2.1. BMO functions A function f ∈ L1loc (Rn ) if called of bounded mean oscillation if ∫ 1 |f (y) − fB(x,r) | dy < ∞, ∥f ∥BM O := sup x∈Rn ,r>0 |B(x, r)| B(x,r) ∫ 1 f (y) dy is the mean of f over the ball B. The space BM O(Rn ), collecting all those where fB := |B| B functions f on Rn such that ∥f ∥BM O < ∞, is a Banach space (modulo constants) with respect to the norm ∥ · ∥BM O . The next lemma collects some properties of BMO functions which will be useful later on. Lemma 2.1. Let f ∈ BM O(Rn ). (i) For all 1 < p < ∞ we have ( sup x∈Rn ,r>0

1 |B(x, r)|

) p1

∫

p

|f (y) − fB(x,r) | dy

≈ ∥f ∥BM O .

(2.1)

B(x,r)

(ii) For all balls B1 , B2 in Rn such that B1 ⊂ B2 , there holds |fB1 − fB2 | ≲

|B2 | ∥f ∥BM O . |B1 |

(2.2)

(iii) For δ > 1 we have |fB(x,δr) − fB(x,r) | ≲ log(δ + 1) ∥f ∥BM O ,

(2.3)

with the implicit constant independent of f , x, r and δ. Further details and proofs can be found in [16, Chapter 7]. 2.2. Commutators The commutator of a sublinear operator T with a function b ∈ L1loc (Rn ) is defined by [b, T ]f := bT (f ) − T (bf ). It is known that commutators are important tools in some applications, including the study of partial differential equations and Jacobians (cf. [10,16,24,33]). There are many papers in the literature dealing with commutators of classical operators in various function spaces. For instance, in the case of the Calder´on– Zygmund singular integral operator Kf = p.v.(k ∗ f ) (with the kernel k(x) satisfying standard conditions), the celebrated result by Coifman, Rochberg and Weiss [11] states that [b, K] is bounded on Lp (Rn ) when

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1 < p < ∞ and b ∈ BM O(Rn ). Janson [20] proved that b ∈ BM O(Rn ) is also necessary for the Lp boundedness of K. On the other hand, commutators also provide characterizations for the BM O norm itself. In this paper we consider the maximal commutator ∫ 1 |b(x) − b(y)| |f (y)| dy Mb f (x) := sup t>0 |B(x, t)| B(x,t) and also the fractional maximal commutator Mα b f (x) := sup t>0

∫

1

|b(x) − b(y)| |f (y)| dy,

1− α n

|B(x, t)|

0 < α < n.

B(x,t)

For some appropriate functions b both the commutators [b, M ] and [b, M α ], of the Hardy–Littlewood maximal operator ∫ 1 M f (x) := sup |f (y)| dy, (2.4) t>0 |B(x, t)| B(x,t) and the fractional maximal operator M α f (x) := sup t>0

∫

1 |B(x, t)|

1− α n

|f (y)|dy, B(x,t)

respectively, can be controlled by the corresponding maximal commutators. Indeed, if b is non-negative we have |[b, M ]f (x)| ≤ Mb f (x)

and

|[b, M α ]f (x)| ≤ Mα b f (x),

(2.5)

for all f ∈ L1loc (Rn ). Maximal commutators were also used in [14] to control commutators of certain singular integrals. If b ∈ BM O(Rn ) then Mb is known to be bounded on Lp (Rn ) (cf. [14,38]). Moreover, Mb is also bounded on the Morrey spaces (cf. [17,19]). We shall also consider the commutator [b, I α ] of the Riesz potential operator ∫ f (y) I α f (x) := n−α dy. n |x − y| R 3. Maximal commutators on vanishing Morrey spaces It is known that the maximal commutator is bounded on Morrey spaces and also on vanishing Morrey spaces at the origin. The results formulated in the next theorem are given in [19, Corollary 6.3] and [12, Corollary 5.8]. Theorem 3.1.

Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) then Mb is bounded on Lp,λ (Rn ) and ∥Mb f ∥p,λ ≲ ∥b∥BM O ∥f ∥p,λ

for all f ∈ Lp,λ (Rn ). Moreover, Mb is also bounded from V0 Lp,λ (Rn ) into itself. In this section we study the boundedness of Mb on the different vanishing spaces V∞ Lp,λ (Rn ) and V Lp,λ (Rn ). (∗)

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3.1. Preservation of the property (V∞ ) The next result is contained in [12, Lemma 5.5] (see also [19, Lemma 6.3]). Let 1 < p < ∞ and b ∈ BM O(Rn ). Then

Lemma 3.2.

  Mb f  p L (B(x

0 ,r))

(  n t ) − np  f  p ≲ ∥b∥BM O r p sup 1 + ln t L (B(x0 ,t)) r t>r

(3.1)

for all f ∈ Lploc (Rn ) and any ball B(x0 , r). This auxiliary lemma allows us to establish the invariance of V∞ Lp,λ (Rn ) with respect to Mb . Theorem 3.3. into itself.

Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) then Mb is bounded from V∞ Lp,λ (Rn )

Proof . Since the Morrey norm inequalities are already known (cf. Theorem 3.1), it remains to show that V∞ Lp,λ (Rn ) is invariant with respect to Mb : lim sup Mp,λ (f ; x, r) = 0

r→∞ x∈Rn

=⇒

lim sup Mp,λ (Mb f ; x, r) = 0.

r→∞ x∈Rn

If f ∈ V∞ Lp,λ (Rn ) then for any ε > 0 there exists R = R(ε) > 0 such that sup Mp,λ (f ; x, t) < ε for every t ≥ R.

x∈Rn

Using inequality (3.1) in modular form and taking into account the boundedness of the function s ↦→ in (1, ∞), we get

(1+ln s)p sn−λ

( t )p ( t )λ−n Mp,λ (Mb f ; x, r) ≲ ∥b∥pBM O sup 1 + ln Mp,λ (f ; x, t) ≲ ε ∥b∥pBM O r r t>r for any x ∈ Rn and every r ≥ R (with the implicit constants independent of x and r). Therefore lim sup Mp,λ (Mb f ; x, r) = 0

r→∞ x∈Rn

and hence Mb f ∈ V∞ Lp,λ (Rn ). □ In view of (2.5) we obtain the following result for the commutator of the maximal operator. Corollary 3.4. Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) is non-negative then [b, M ] is bounded on V∞ Lp,λ (Rn ). 3.2. Preservation of the property (V ∗ ) As we can see below, the proof of the preservation of (V ∗ ) is much harder. Note that inequalities like (3.1) are mainly useful for studying local integrability on balls related to the behavior of their radius. Theorem 3.5. into itself.

Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) then Mb is bounded from V (∗) Lp,λ (Rn )

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Proof . Recalling the boundedness of Mb on Lp,λ (Rn ), we have to show now that lim AN,p (f ) = 0

N →∞

( ) lim AN,p Mb f = 0.

=⇒

N →∞

Step 1 : For x ∈ Rn and N ∈ N, we split f into f = f1 + f2 ,

with f1 := f χΩx,N/2 ,

f2 = f χRn \Ωx,N/2 ,

using the notation ( ) Ωx,M := B(x, 2) ∩ Rn \ B(0, M ) , M > 0. Since Mb is sublinear, we have ( ) ( ) ( ) AN,p Mb f ≲ AN,p Mb (f1 ) + AN,p Mb (f2 ) .

(3.2)

We show next that both quantities on the right-hand side of (3.2) tend to zero as N → ∞. The boundedness of Mb on Lp (Rn ) gives ∫ ∫ ∫ ∫ ( )p ( )p p p Mb (f1 )(y) χN (y) dy ≤ Mb (f1 )(y) dy ≲ |f1 (y)| dy = |f (y)| dy Rn

B(x,1)

Rn

Ωx,N/2

with the implicit constants independent of x, N and f . Since f ∈ V (∗) Lp,λ (Rn ), the right hand side above tends to zero uniformly on x as N → ∞ (note that the property (V ∗ ) does not depend on the particular value ( ) of the radius taken in the balls centered at x, cf. [6, Lemma 3.4]). Therefore, limN →∞ AN,p Mb (f1 ) = 0. Step 2 : Now we deal with the second term in the sum in (3.2). We have ∫ ( )p Mb (f2 )(y) χN (y) dy ≲ I(x, N, t0 ) + II(x, N, t0 ) B(x,1)

with [

∫ I(x, N, t0 ) :=

χN (y) sup B(x,1)

0
]p

∫

1 |B(y, t)|

B(y,t)

|b(y) − b(z)| |f (z)| χRn \Ωx,N/2 (z) dz

dy

and ∫

[

1 II(x, N, t0 ) := χN (y) sup |B(y, t)| t≥t0 B(x,1)

]p

∫ B(y,t)

|b(y) − b(z)| |f (z)| χRn \Ωx,N/2 (z) dz

dy,

where t0 > 0 is a certain fixed number (to be chosen below). Step 3 : We estimate II(x, N, t0 ). Let ε > 0 be arbitrary. We have II(x, N, t0 ) ≤ II1 (x, N, t0 ) + II2 (x, N, t0 ) with ∫

[

1 II1 (x, N, t0 ) := χN (y) sup t≥t0 |B(y, t)| B(x,1)

]p

∫ B(y,t)

|b(y) − bB(y,t) | |f (z)| χRn \Ωx,N/2 (z) dz

dy,

and ∫

[

1 II2 (x, N, t0 ) := χN (y) sup |B(y, t)| t≥t0 B(x,1)

]p

∫ B(y,t)

|b(z) − bB(y,t) | |f (z)| χRn \Ωx,N/2 (z) dz

dy.

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By H¨ older’s inequality and the characterization (2.1) we get ∫ ∫ 1 p p |f (z)| dz dy ≲ ∥b∥pBM O ∥f ∥pp,λ ε, II2 (x, N, t0 ) ≤ ∥b∥BM O χN (y) sup t≥t0 |B(y, t)| B(y,t) B(x,1)

(3.3)

after choosing t0 > 1 large enough so that tλ−n < ε for all t ≥ t0 (recall that λ < n). Now we deal with the quantity II1 (x, N, t0 ). Splitting |b(y) − bB(y,t) | ≤ |b(y) − bB(x,1) | + |bB(x,1) − bB(y,2) | + |bB(y,2) − bB(y,t) |, in the integral in II1 (x, N, t0 ), we get II1 (x, N, t0 ) ≲ ∥b∥pBM O ∥f ∥pp,λ ε + ∥f ∥pp,λ

∫

p

sup |bB(y,2) − bB(y,t) | tλ−n dy

B(x,1) t≥t0

as above. Since B(y, 2) ⊂ B(y, t) for all t ≥ t0 ≥ 2, by property (2.3) we have p

|bB(y,2) − bB(y,t) | tλ−n ≲ logp (1 + t/2) ∥b∥pBM O tλ−n ≤ ∥b∥pBM O ε in the last integral, if we take, in addition, t0 large enough such that logp (1 + t/2) tλ−n ≤ ε for all t ≥ t0 . Therefore, II1 (x, N, t0 ) ≲ ∥b∥pBM O ∥f ∥pp,λ ε. Combining this with (3.3), we see that II(x, N, t0 ) ≲ ∥b∥pBM O ∥f ∥pp,λ ε ,

uniformly on x ∈ Rn (and N ),

for a convenient choice of t0 . Step 4 : In this step we deal with I(x, N, t0 ). There are two different cases to be considered when z ∈ / Ωx,N/2 (and z ∈ B(y, t)). If z ∈ B(0, N/2) then t > |z − y| ≥ |y| − |z| > N/2. Thus there is no contribution to the supremum on t ∈ (0, t0 ) for N ≥ 2t0 . If z ∈ / B(x, 2) then t > |z − y| ≥ |z − x| − |y − x| ≥ 1. Hence it remains to handle I(x, N, t0 ) when the supremum is taken over all t ∈ (1, t0 ). Then we have [∫ ]p ∫ I1 (x, N, t0 ) ≲

|b(y) − b(z)| |f (z)| dz

χN (y) B(x,1)

dy

B(y,t0 )

=: I1 (x, N, t0 ) + I2 (x, N, t0 ) + I3 (x, N, t0 ), where

]p

[∫

∫ I1 (x, N, t0 ) :=

|b(y) − bB(x,1) | |f (z)| dz

χN (y) B(x,1)

dy,

B(y,t0 )

]p

[∫

∫ I2 (x, N, t0 ) :=

χN (y) B(x,1)

B(y,t0 )

χN (y) B(x,1)

B(y,t0 )

dy,

]p

[∫

∫ I3 (x, N, t0 ) :=

|bB(x,1) − bB(y,t0 ) | |f (z)| dz |bB(y,t0 ) − b(z)| |f (z)| dz

In I3 (x, N, t0 ) we apply H¨ older’s inequality of exponent s ∈ (1, p) and get ∫   |bB(y,t0 ) − b(z)| |f (z)| dz ≤ bB(y,t0 ) − b(·)Ls′ (B(y,t

0 ))

B(y,t0 )

n/s′

≲ ∥b∥BM O t0

∥f ∥Ls (B(y,t0 ))

∥f ∥Ls (B(y,t0 ))

≈ ∥b∥BM O ∥f ∥Ls (B(y,t0 ))

dy.

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(after using (2.1) in the last inequality). A change of variables and Minkowski’s inequality of exponent p/s yield [∫ ] ps ∫ s p I3 (x, N, t0 ) ≲ ∥b∥BM O χN (y) χB(0,t0 ) (z) |f (y − z)| dz dy Rn

B(x,1)

≤ ∥b∥pBM O

⎧ ⎨∫ ⎩

≤ ∥b∥pBM O

Rn

⎧ ⎨∫ ⎩

Rn

] ps

[∫

p

B(x,1)

χN (y) χB(0,t0 ) (z) |f (y − z)| dy

dz

⎫ ps ⎬ ⎭

[

] ps

∫

χB(0,t0 ) (z) sup

v∈Rn

p

χN −|z| (u) |f (u)| du

dz

B(v,1)

⎫ ps ⎬ ⎭

with the interpretation χa := 1

if a ≤ 0

and χa := χRn \B(0,a) if a > 0.

(3.4)

Therefore, taking also into account that f ∈ Lp,λ (Rn ), we conclude that I3 (x, N, t0 ) → 0

as N → ∞ (uniformly on x ∈ Rn )

by the Lebesgue convergence theorem. To show that I2 (x, N, t0 ) → 0 as N → ∞ (also uniformly on x ∈ Rn ), first we note that |bB(x,1) − bB(y,t0 ) | ≤ tn0 ∥b∥BM O , since B(x, 1) ⊂ B(y, t0 ) for y ∈ B(x, 1) and t0 ≥ 2, and then apply a change of variables and Minkowski’s inequality (now with exponent p) as in I3 (x, N, t0 ). Finally we deal with I1 (x, N, t0 ). In this case we have χN (y) ≤ χN −t0 (z) (again with the interpretation (3.4)) for all z ∈ B(y, t0 ), with |y| > N . Moreover, B(y, t0 ) ⊂ B(x, 2t0 ) for y ∈ B(x, 1). Therefore, applying also H¨ older’s inequality, we get [∫ ]p ∫ p I1 (x, N, t0 ) ≤ |b(y) − bB(x,1) | χN −t0 (z) |f (z)| dz dy B(y,t )

B(x,1)

0 ∫ p  p b(·) − bB(x,1)  p χN −t0 (z) |f (z)| dz ≤ L (B(x,1)) B(x,2t0 ) B(x,1) ∫ p p ≈ ∥b∥BM O χN −t0 (z) |f (z)| dz.

∫

B(x,2t0 )

Since the vanishing property (V ∗ ) does not depend on the size of the radius we set in the ball centered at x (cf. [6, Lemma 3.4]), we have I1 (x, N, t0 ) ≲ ∥b∥pBM O AN −t0 ,p (f ) (recall that t0 is fixed), from which we get as N → ∞ (uniformly on x ∈ Rn ). ( ) Conjugating all the estimates obtained in Steps 2–4, we see that AN,p Mb (f2 ) → 0 as N → ∞. Now combining this with the conclusion in Step 1, by (3.2) we finally conclude that ( ) lim AN,p Mb (f ) = 0. I1 (x, N, t0 ) → 0

N →∞

The proof is complete.

□

In view of (2.5) we also have the following result: Corollary 3.6. Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) is non-negative then [b, M ] is bounded on V (∗) Lp,λ (Rn ).

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4. Fractional maximal commutators and commutators of the riesz potential operator ( ) The following theorem gives an Adams type result on the Lp,λ → Lq,λ - boundedness of the fractional maximal operator, cf. [19, Theorem 6.10, Corollary 6.4]. Recall that the action of fractional integral operators on Morrey spaces goes back to Adams [1]. Theorem 4.1. Let 0 < α < n, 0 ≤ λ < n, 1 < p < (n − λ)/α and 1/q = 1/p − α/(n − λ). Then Mα b is p,λ n q,λ n bounded from L (R ) into L (R ). This can be proved from the inequality below which shows that the fractional maximal commutator with BM O coefficients can be pointwisely estimated by the maximal commutator of Morrey functions. A proof of (4.1) can be found in [19, p. 189]. Lemma 4.2. Let α, λ, p, q be as in Theorem 4.1. For any f ∈ Lp,λ (Rn ) we have p ( ) pq 1− q ∥f ∥ |Mα f (x)| ≲ ∥b∥ M f (x) BM O b b p,λ

(4.1)

for r > 0 and x ∈ Rn . The next result shows that the commutator Mα b preserves all the vanishing properties (V0 ), (V∞ ) and (V ). ∗

Theorem 4.3. Let 0 < α < n, 0 ≤ λ < n, 1 < p < (n − λ)/α and 1/q = 1/p − α/(n − λ). If b ∈ BM O(Rn ), p,λ then the commutator Mα (Rn ) into V0 Lq,λ (Rn ), from V∞ Lp,λ (Rn ) into V∞ Lq,λ (Rn ) b is bounded from V0 L (∗) p,λ n (∗) q,λ n and from V L (R ) into V L (R ). Proof . The norm inequalities are clear from Theorem 4.1. To show the preservation of the vanishing properties we make use of the inequality (4.1). Then we get ( ) ( ) q q−p Mq,λ Mα b f ; x, r ≲ ∥b∥BM O ∥f ∥p,λ Mp,λ Mb f ; x, r for any r > 0 and any x ∈ Rn . Hence, if f ∈ V∞ Lp,λ (Rn ) then Mb f ∈ V∞ Lp,λ (Rn ) by Theorem 3.3. q,λ Consequently, we have Mα (Rn ). The corresponding boundedness on V0 Lp,λ (Rn ) can be obtained b f ∈ V∞ L using similar arguments, observing that Mb f also preserves (V0 ), see [12, Theorem 5.7, Corollary 5.8]. As regards the vanishing property (V ∗ ), by (4.1) we also get ( ) q q−p AN,q Mα b f ≲ ∥b∥BM O ∥f ∥p,λ AN,p (Mb f ) with the implicit constant independent of f and N ∈ N. If f ∈ V (∗) Lp,λ (Rn ) then Mb f ∈ V (∗) Lp,λ (Rn ) by (∗) q,λ Theorem 3.5. Consequently, we have Mα L (Rn ). □ bf ∈V ( Next we discuss the action of commutators of Riesz potentials on vanishing Morrey spaces. The Lp → ) Lq - boundedness of commutators of fractional integral operators goes back to Chanillo [8], who proved that [b, I α ] is bounded from Lp (Rn ) into Lq (Rn ), with 0 < α < n, 1 < p < n/α, 1/q = 1/p − α/n, if and only if b ∈ BM O(Rn ). The study of such commutators in classical Morrey spaces was performed by Di Fazio and Ragusa [13] and later by Komori and Mizuhara [22] and Shirai [39], where it was observed that b ∈ BM O(Rn ) is also necessary and sufficient to the boundedness of [b, I α ] from Lp,λ (Rn ) into Lq,λ (Rn ), where 0 < α < n, 1 < p < ∞, 0 < λ < n − αp, 1/q = 1/p − α/(n − λ). Additional results of Spanne type can be found in [19]. See also [4] for regularity of commutators in Morrey spaces.

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As shown in [19, p. 193–194], the commutator of Riesz potentials of Morrey functions can be controlled by the maximal commutator like the fractional maximal commutator in (4.1): p ( )p 1− |[b, I α ]f (x)| ≲ ∥b∥BM O Mb f (x) q ∥f ∥p,λ q

(4.2)

with the implicit constant independent of f ∈ Lp,λ (Rn ), b ∈ BM O(Rn ), r > 0 and x ∈ Rn . Hence, as in the case of the operator Mα b , we can formulate the following statement: Theorem 4.4. Let 0 < α < n, 0 ≤ λ < n, 1 < p < (n − λ)/α and 1/q = 1/p − α/(n − λ). If b ∈ BM O(Rn ), then the commutator [b, I α ] is bounded from V0 Lp,λ (Rn ) into V0 Lq,λ (Rn ), from V∞ Lp,λ (Rn ) into V∞ Lq,λ (Rn ) and from V (∗) Lp,λ (Rn ) into V (∗) Lq,λ (Rn ). 5. Commutators on the closure of C0∞ (Rn ) It is known that Lp,λ (Rn ) is non-separable when 0 < λ ≤ n (an explicit proof can be found in [35]). Let ◦

us denote by Lp,λ (Rn ) the closure of C0∞ (Rn ) in the norm ∥ · ∥p,λ . In [6], it was shown that such closure can be explicitly described by the intersection of the three vanishing subspaces: ◦

(∗)

Lp,λ (Rn ) = V0,∞ Lp,λ (Rn ) := V0 Lp,λ (Rn ) ∩ V∞ Lp,λ (Rn ) ∩ V (∗) Lp,λ (Rn ) (see also [46]). This closure plays an important role in harmonic analysis on Morrey spaces since C0∞ (Rn ) is (◦ )′ dense in Lp,λ (Rn ) and, moreover, its dual provides a predual space for Morrey spaces: )′ ) ′ (◦ )′′ (( ◦ Lp,λ (Rn ) Lp,λ (Rn ) = = Lp,λ (Rn ) (cf. [2,3,35,43]). Notice also that the strict embeddings (∗)

V0,∞ Lp,λ (Rn ) ⫋ V0 Lp,λ (Rn ) ∩ V∞ Lp,λ (Rn ) ⫋ V0 Lp,λ (Rn ) ⫋ Lp,λ (Rn ) hold (cf. [6]). Using the results from Sections 3 and 4, and also the already known boundedness results on the vanishing spaces at the origin, we get the following corollaries on the action of maximal commutators on the closure of C0∞ (Rn ). ◦

Corollary 5.1. Let 1 < p < ∞ and 0 ≤ λ < n. If b ∈ BM O(Rn ) then Mb is bounded from Lp,λ (Rn ) into itself. Corollary 5.2. Let 0 < α < n, 0 ≤ λ < n, 1 < p < (n − λ)/α and 1/q = 1/p − α/(n − λ). If b ∈ BM O(Rn ) ◦

◦

p,λ then Mα (Rn ) into Lq,λ (Rn ). b is bounded from L

Acknowledgments This research was partially supported by FCT, Portugal through CIDMA — Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2019.

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