Some techniques on nonlinear analysis and applications

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Advances in Mathematics 229 (2012) 1235–1265 www.elsevier.com/locate/aim

Some techniques on nonlinear analysis and applications Daniel Pellegrino a,1,∗ , Joedson Santos b , Juan B. Seoane-Sepúlveda c,2 a Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900 – João Pessoa, Brazil b Departamento de Matemática, Universidade Federal de Sergipe, 49.500-000 – Itabaiana, Brazil c Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, Universidad Complutense de Madrid,

Plaza de las Ciencias 3, 28040 Madrid, Spain Received 28 October 2010; accepted 29 September 2011 Available online 8 November 2011 Communicated by R. Daniel Mauldin

Abstract In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second result, more technical, but also connected to the first one, is an extension of the well known Pietsch Domination Theorem. The last decade witnessed the birth of different families of Pietsch Domination-type results and some attempts of unification. Our result, that we call “full general Pietsch Domination Theorem” is potentially a definitive Pietsch Domination Theorem which unifies the previous versions and delimits what can be proved in this line. The connections to the recent notion of weighted summability are traced. © 2011 Elsevier Inc. All rights reserved. Keywords: Absolutely summing operators; Absolutely summing multilinear maps; Inclusion Theorem; Pietsch Domination Theorem

* Corresponding author.

E-mail addresses: [email protected] (D. Pellegrino), [email protected] (J. Santos), [email protected] (J.B. Seoane-Sepúlveda). 1 The author was supported by CNPq Grant 301237/2009-3. 2 The author was supported by the Spanish Ministry of Science and Innovation, grant MTM2009-07848. 0001-8708/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2011.09.014

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1. Introduction and motivation Common, even simple, mathematical problems usually involve nonlinear maps, sometimes acting on sets with little (or none) algebraic structure; so the extension of linear techniques to the nonlinear setting, besides its intrinsic mathematical interest, is an important task for potential applications. In fact it is mostly a challenging task, since linear arguments are commonly ineffective in a more general setting. The following problem illustrates this situation. If X, Y are Banach spaces, u, v : X → Y are continuous linear operators, C > 0 and 1  p  q < ∞ it is possible to show that: 1. If m m       u(xj )p  C v(xj )p j =1

for every m and all x1 , . . . , xm ∈ X,

(1.1)

for every m and all x1 , . . . , xm ∈ X.

(1.2)

j =1

then m m       u(xj )q  C v(xj )q j =1

j =1

Also, in the same direction: 2. If m m       u(xj )p  C sup ϕ(xj )p j =1

ϕ∈BX∗ j =1

for every m and all x1 , . . . , xm ∈ X,

(1.3)

for every m and all x1 , . . . , xm ∈ X,

(1.4)

then m m       u(xj )q  C sup ϕ(xj )q j =1

ϕ∈BX∗ j =1

where X ∗ is the topological dual of X and BX∗ denotes its closed unit ball. More generally, if pj  q j

for j = 1, 2,

1  p1  p2 < ∞, 1  q1  q2 < ∞, 1 1 1 1 −  − , p1 q 1 p2 q 2 then

(1.5)

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m    u(xj )q1

1/q1

  C sup

ϕ∈BX∗

j =1

m    ϕ(xj )p1

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1/p1 (1.6)

j =1

for every m and all x1 , . . . , xm ∈ X, implies that 

m    u(xj )q2

1/q2

  C sup

ϕ∈BX∗

j =1

m    ϕ(xj )p2

1/p2 (1.7)

j =1

for every m and all x1 , . . . , xm ∈ X. Problem 1.1. What about nonlinear versions of the above results? Are there any? Problem 1.2. What about nonlinear versions in which the spaces X and Y are just sets, with no structure at all? The interested reader can find the proof of the implication (1.6) ⇒ (1.7) in [27, p. 198]. This result was essentially proved by S. Kwapie´n in 1968 (see [42]) and it is what is now called “Inclusion Theorem for absolutely summing operators”. A quick look shows that the linearity is fully explored and a nonlinear version of this result, if there is any, would require a whole new technique. It is worth mentioning that practical problems may also involve sets with less structure than Banach spaces (or less structure than linear spaces or even than metric spaces) and a “full” nonlinear version (with no structure on the spaces involved) would certainly be interesting for potential applications. In this direction we will prove a very general result, which we will call “Inclusion Principle”, which, due its extreme generality, may be useful in different contexts, even outside of pure mathematical analysis. The arguments used in the proof of the “Inclusion Principle” are, albeit tricky, fairly clear and simple in nature, but we do believe this technique may be useful in different contexts. To illustrate its reach, at least in the context of Functional Analysis, we show that very particular cases of the Inclusion Principle can contribute to the nonlinear theory of absolutely summing operators. Below, as an illustration, we describe an extremely particular case of the forthcoming Inclusion Principle: Let X be an arbitrary non-void set and Y be a normed space; suppose that pj and qj satisfy (1.5). If f, g : X → Y are arbitrary mappings and there is a constant C > 0 so that m m       f (xj )q1  C g(xj )p1 , j =1

j =1

for every m and all x1 , . . . , xm ∈ X, then there is a constant C1 > 0 such that 

m    f (xj )q2 j =1

1

α

 C1

m    g(xj )p2 j =1

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for every m and all x1 , . . . , xm ∈ X, with α=

q 2 p1 q 1 p2

if p1 < p2 .

The case p1 = p2 is trivial. The parameter α is a kind of adjustment, i.e., the price that one has to pay for the complete lack of linearity, and precisely when pj = qj for j = 1 and 2 we have α = 1 and no adjustment is needed. In other words, the parameter α indicates the necessary adjustments (in view of the lack of linearity) when pj and qj become distant. The second main contribution of this paper is more technical, but also useful. It is what we call “full general Pietsch Domination Theorem” which, as will be shown, has several applications and seems to be a definitive answer to the attempt of delimiting the amplitude of action of Pietsch Domination-type theorems. The Pietsch Domination Theorem (PDT) (sometimes stated as the Pietsch Factorization Theorem) was proved in 1967 by A. Pietsch, in his classical paper [65], and since then it has played a special and important role in Banach Space Theory having its honour place in several textbooks related to Banach Space Theory [3,21,27,66,70,72]; PDT has a strong connection with the aforementioned inclusion results, as we explain below. In fact, if 0 < p < ∞, PDT states that for a given continuous linear operator u : X → Y the following assertions are equivalent: (i) There exists a C > 0 so that m m       u(xj )p  C sup ϕ(xj )p j =1

for every m.

ϕ∈BX∗ j =1

(ii) There are a Borel probability measure μ on BX∗ (with the weak-star topology) and C > 0 such that   u(x)  C



  ϕ(x)p dμ

1

p

.

(1.8)

BX∗

Using the canonical inclusions between Lp spaces we conclude that, if 0 < p  q < ∞, the inequality (1.8) implies that   u(x)  C



  ϕ(x)q dμ

1 q

BX∗

and we obtain the implication (1.3) ⇒ (1.4) as a corollary. Due to its strong importance in Banach Space Theory, PDT was re-discovered in different contexts in the last decades (e.g. [1,19,28,32,33,47,48,56]) and, since 2009, in [16,17,58] some attempts were made in the direction of showing that one unique PDT can be stated in such a general way that all the possible Pietsch Domination-type theorems would be straightforward particular cases of this unified Pietsch Domination Theorem. Thus, the second contribution of this paper is to prove a “full general Pietsch Domination Theorem” that, besides its own interest, we do believe that will be useful to delimit the scope of

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Pietsch-type theorems. Some connections with the recent promising notion of weighted summability introduced in [59] are traced. 2. The Inclusion Principle In this section we deal with general values for pj and qj satisfying (1.5). In order to be useful in different contexts, we state the result in a very general form. Let X, Y , Z, V and W be (arbitrary) non-void sets. The set of all mappings from X to Y will be represented by Map(X, Y ). Let H ⊂ Map(X, Y ) and R : Z × W → [0, ∞),

and

S : H × Z × V → [0, ∞) be arbitrary mappings. If 1  p  q < ∞, suppose that sup

m 

R(zj , w)p < ∞

and

w∈W j =1

sup

m 

S(f, zj , v)q < ∞

v∈V j =1

for every positive integer m and z1 , . . . , zm ∈ Z (in most of the applications V and W are compact spaces and R and S have some trace of continuity to assure that both sup are finite). If α ∈ R, we will say that f ∈ H is RS-abstract ((q, α), p)-summing (notation f ∈ RS((q,α),p) ) if there is a constant C > 0 so that  sup

m 

1

α

q

S(f, zj , v)

v∈V j =1

 C sup

m 

R(zj , w)p ,

(2.1)

w∈W j =1

for all z1 , . . . , zm ∈ Z and m. Theorem 2.1 (Inclusion Principle). If pj and qj satisfy (1.5), then RS((q1 ,1),p1 ) ⊂ RS((q2 ,α),p2 ) for α=

q 2 p1 q 1 p2

if p1 < p2 .

Proof. Let f ∈ RS((q1 ,1),p1 ) . There is a C > 0 such that sup

m 

v∈V j =1

S(f, zj , v)q1  C sup

m 

R(zj , w)p1 ,

(2.2)

w∈W j =1

for all z1 , . . . , zm ∈ Z and m ∈ N. If each η1 , . . . , ηm is a positive integer, by considering each zj repeated ηj times in (2.2) one can easily note that

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sup

m 

ηj S(f, zj , v)q1  C sup

v∈V j =1

m 

ηj R(zj , w)p1 ,

(2.3)

w∈W j =1

for all z1 , . . . , zm ∈ Z and m ∈ N. Now, using a clever argument credited to Mendel and Schechtman (used recently, in different contexts, in [32,59,58]) we can conclude that (2.3) holds for arbitrary positive real numbers ηj . The idea is to pass from integers to rationals by “cleaning” denominators and from rationals to real numbers using density. It is worth mentioning that this technique is, in essence, much older and relies on the arguments that show that a continuous solution to the functional equation f (x + y) = f (x) + f (y) must be linear. Since p1 < p2 we have q1 < q2 . Define p, q as 1 1 1 = − p p1 p2

1 1 1 = − . q q1 q2

and

So we have 1  q  p < ∞; next, let m ∈ N and z1 , z2 , . . . , zm ∈ Z be fixed. For each j = 1, . . . , m, consider the map λj : V → [0, ∞), q2

λj (v) := S(f, zj , v) q . Thus, λj (v)q1 S(f, zj , v)q1 = S(f, zj , v)

q1 q2 q

S(f, zj , v)q1

= S(f, zj , v)q2 . Recalling that (2.3) is valid for arbitrary positive real numbers ηj , we get, for ηj = λj (v)q1 , m 

S(f, zj , v)q2 =

j =1

m 

λj (v)q1 S(f, zj , v)q1

j =1

 C sup

m 

λj (v)q1 R(zj , w)p1

w∈W j =1

for every v ∈ V . Also, since p, p2 > p1 and we obtain m 

S(f, zj , v)q2  C sup

m 

1 (p/p1 )

+

= 1, invoking Hölder’s Inequality

1 (p2 /p1 )

λj (v)q1 R(zj , w)p1

w∈W j =1

j =1

  C sup w∈W

 =C

m  j =1

m 

λj (v)

q1 p p1

 p1  p

j =1

λj (v)

q1 p p1

m 

 p1  R(zj , w)p2

p2

j =1

 p1



p

sup w∈W

m  j =1

 p1 R(zj , w)p2

p2

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for every v ∈ V . Since m 

q1 p p1

1241

 p  q we have . q1 p  .q and then p1

 S(f, zj , v)  C q2

j =1

m 

 q1 sup

λj (v)

w∈W

j =1

 =C

m 



q

q



m 

 sup w∈W

j =1

p2

R(zj , w)

j =1

q1 q

S(f, zj , v)q2

 p1 p2

m 

 p1 R(zj , w)p2

p2

j =1

for every v ∈ V . We thus have 

m 

1− q1



q

 C sup

S(f, zj , v)q2

w∈W

j =1

m 

 p1 R(zj , w)p2

p2

j =1

for every v ∈ V , and we can finally conclude that  sup

m 

v∈V j =1

 q1 p2 S(f, zj , v)q2

q2 p1

p2

 C p1 sup

m 

R(zj , w)p2 .

2

w∈W j =1

Remark 2.2. It is interesting to mention that as qj becomes closer to pj for j = 1 and 2, the value qq12 pp21 becomes closer to 1 (which occurs in the linear setting when pj = qj for j = 1 and 2). In other words, the effect of the lack of linearity in our estimates is weaker when pj and qj are closer and, in the extreme case where p1 = q1 and p2 = q2 , then α = 1 and we have a “perfect generalization” of the linear result. 3. Applications of the Inclusion Principle to classes of nonlinear absolutely summing operators 3.1. Absolutely summing linear and multilinear operators: a brief summary In the real line it is well known that a series is absolutely convergent precisely when it is unconditionally convergent. For infinite-dimensional Banach spaces it is easy to verify that the situation is different; for example, for p spaces with 1 < p < ∞, it is easy to construct an unconditionally convergent series which fails to be absolutely convergent. However the behavior for arbitrary Banach spaces was not known before 1950. For 1 , for example, the construction is much more complicated (see M.S. McPhail’s work from 1947 [46]). This perspective leads to the feeling that this property (having an unconditionally summable series which is not absolutely summable) could be shared by all infinite-dimensional Banach spaces. This question was raised by S. Banach in his monograph [5, p. 40] and appears as Problem 122 in the Scottish Book (see [52]). In 1950, A. Dvoretzky and C.A. Rogers [31] solved this question by showing that in every infinite-dimensional Banach space there is an unconditionally convergent series which fails to be absolutely convergent. This new panorama of the subject attracted the attention of A. Grothendieck who provided, in his thesis [35], a different approach to the Dvoretzky–Rogers result. His thesis, together with his Résumé [34], can be regarded as the beginning of the theory

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of absolutely (q, p)-summing operators. Grothendiek’s version of Dvoretzky–Rogers theorem is the following lemma (for further details we refer to [26]): Lemma 3.1 (Dvoretzky–Rogers Theorem—Grothendiek’s version). Let X be an n-dimensional Banach space. There exist norm-one vectors x1 , . . . , xn in X such that if 1  r  n, then for any λ1 , . . . , λr ∈ R,   r       λi xi   Mr (λi )2 ,  r   i=1

where Mr = 1 +

√

1/2 1 2 r r 1 + 22 + · · · + (r − 1)2 1+ √ . n n 3

The notion of absolutely (q, p)-summing operator, as we know nowadays, is due to B. Mitiagin and A. Pełczy´nski [54] and A. Pietsch [65]. Pietsch’s paper is a classic and a pivotal role is played by the Domination Theorem, which presents an unexpected measure-theoretical characterization of p-summing operators. The same task was brilliantly done, one year later, by J. Lindenstrauss and A. Pełczy´nski’s paper [44] which reformulated Grothendieck’s tensorial arguments giving birth to a comprehensible theory with broad applications in Banach Space Theory. From now on the space of all continuous linear operators from a Banach space X to a Banach space Y will be denoted by L(X, Y ). If 1  p  q < ∞, we say that the Banach space operator u : X → Y is (q, p)-summing if there is an induced operator strong

(X) → q uˆ : weak p

(Y )

∞ (xn )∞ n=1 → (uxn )n=1 .

 ∞ p 1/p < ∞}. The Above weak (X) := {(xj )∞ p j =1 ⊂ X : (xj )j =1 w,p := supϕ∈BX∗ ( j |ϕ(xj )| ) class of absolutely (q, p)-summing linear operators from X to Y will be represented by Πq,p (X, Y ). For details on the linear theory of absolutely summing operators we refer to the classical book [27]. The linear theory of absolutely summing operators was intensively investigated in the 70’s and several classical papers can tell the story (we mention [6,7,20,25,30,53] and the monograph [27] for a complete panorama). A special role is played by Grothendieck’s Theorem and the Pietsch Domination Theorem: Theorem 3.2 (Grothendieck). Every continuous linear operator from 1 to 2 is absolutely (1, 1)summing. Theorem 3.3 (Lindenstrauss and Pełczy´nski). If X and Y are infinite-dimensional Banach spaces, X has an unconditional Schauder basis and Π1,1 (X, Y ) = L(X, Y ) then X = 1 and Y is a Hilbert space. Theorem 3.4 (Pietsch Domination Theorem). If X and Y are Banach spaces, a continuous linear operator T : X → Y is absolutely (p, p)-summing if and only if there is a constant C > 0 and

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a Borel probability measure μ on the closed unit ball of the dual of X, (BX∗ , σ (X ∗ , X)), such that   T (x)  C



  ϕ(x)p dμ

1

p

.

BX∗

An immediate consequence of the Pietsch Domination Theorem is that, for 1  r  s < ∞, every absolutely (r, r)-summing operator is absolutely (s, s)-summing. However a more general result is valid. As mentioned in the first section, this result is essentially due to Kwapie´n [42]: Theorem 3.5 (Inclusion Theorem). If X and Y are Banach spaces and pj  q j

for j = 1, 2,

1  p1  p2 < ∞, 1  q1  q2 < ∞, 1 1 1 1 −  − , p1 q 1 p2 q 2

(3.1)

then Πq1 ,p1 (X, Y ) ⊂ Πq2 ,p2 (X, Y ).

(3.2)

The early 70’s, more precisely 1972, can be probably recalled as the time of birth of the notion of cotype. A special role was played by the works of S. Kwapie´n [43], H. Rosenthal [69] and Dubinsky, Pełczy´nski and Rosenthal [30]. The term “cotype” appeared in 1974 with the work of J. Hoffmann-Jørgensen [37]. Two years later, with the seminal paper “Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach” by B. Maurey and G. Pisier [53], the connection of the notion of cotype and the concept of absolutely summing operators became clear. For more historical details we refer to [68, pp. 314–315]. In 1992 M. Talagrand [71] proved very deep results complementing previous results of Maurey and Pisier showing that cotype 2 spaces occupy a special place in the theory of absolutely summing operators: Theorem 3.6 (Maurey, Pisier and Talagrand). If a Banach space X has cotype q, then idX is absolutely (q, 1)-summing. The converse is true, except for q = 2. In the last two decades the interest of the theory was moved to the nonlinear setting although there are still some challenging questions being investigated in the linear setting (see [12,18,40, 41,57]). For example, recent results from [18] complement the Lindenstrauss–Pełczy´nski Theorem 3.3 (below cot X denotes the infimum of the cotypes assumed by X; it is interesting to mention that X need not to have cotype cot X): Theorem 3.7. (See [18].) Let X and Y be infinite-dimensional Banach spaces. (i) If Π1,1 (X, Y ) = L(X, Y ) then cot X = cot Y = 2. (ii) If 2  r < cot Y and Πq,r (X, Y ) = L(X, Y ), then L(1 , cot Y ) = Πq,r (1 , cot Y ).

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The extension of the classical linear theory of absolutely summing operators to the multilinear setting is very far from being a mere exercise of generalization with expected results obtained by induction. In fact, some multilinear approaches are simple but there are several delicate, surprising and intriguing questions related to the multilinear extensions of absolutely summing operators. Indeed, let us now take the time to present examples of some surprising/challenging results/questions related to the multilinear setting which justify the efforts of the last years dedicated to the constructions of multilinear “prototypes” of absolutely summing operators. We begin with the class of multiple summing multilinear operators which, so far, is the most famous multilinear extension of the linear concept of absolutely summing operators. Let us recall its definition, which was introduced, independently, by Matos [49] and Bombal, Pérez-García and Villanueva [9,63]: From now on we will use the notation L(X1 , . . . , Xn ; Y ) to represent the spaces of continuous n-linear mappings from X1 × · · · × Xn to Y . If 1  p1 , . . . , pn  q < ∞, we say that a continuous n-linear operator T : X1 × · · · × Xn → Y is multiple (q; p1 , . . . , pn )-summing (T ∈ Lm,(q,p1 ,...,pn ) (X1 , . . . , Xn ; Y )) if there exists a constant C > 0 such that 

∞ 

 (1)

 T x , . . . , x (n) q

j1 ,...,jn =1

j1

jn

1/q C

n   (k) ∞   x  j j =1 w,p

k

(3.3)

k=1

w for every (xj )∞ j =1 ∈ pk (Xk ), k = 1, . . . , n. When p1 = · · · = pn = p, it is commonly written Lm,(q;p) instead of Lm,(q;p1 ,...,pn ) and when p1 = · · · = pn = p = q one can write Lm,p instead of Lm,(p;p,...,p) . The infimum of the constants C satisfying (3.3) defines a norm denoted by πq;p1 ,...,pn (or πq;p if p1 = · · · = pn = p or even πp when p1 = · · · = pn = p = q). The main surprises offered by the class of multiple summing multilinear operators are perhaps related to “inclusion results”. As we have already mentioned it is well known that in the linear theory, when 1  r  s < ∞, every absolutely (r, r)-summing operator is absolutely (s, s)summing. Using techniques of probability in Banach spaces, Pérez-García (2003) proved that the multiple summing version of the aforementioned result holds (in general) just for 1  p  q < 2: (k)

Theorem 3.8 (Pérez-García). (See [62,61].) If 1  p  q < 2, then Lm,p (X1 , . . . , Xn ; Y ) ⊂ Lm,q (X1 , . . . , Xn ; Y ). Moreover, Pérez-García also proved that if 1  p  2 < q there exists T ∈ Lm,p (2 1 ; K)\ Lm,q (2 1 ; K). Some information on the cotype of Y allows a slight improvement: Theorem 3.9 (Pérez-García). (See [62,61].) If 1  p  q  2 and Y has cotype 2, then Lm,p (X1 , . . . , Xn ; Y ) ⊂ Lm,q (X1 , . . . , Xn ; Y ). A completely different technique (complex interpolation + complexification argument) allows to prove a full general version of Theorem 3.8 (with no upper bound for q) when the spaces from the domain are L∞ -spaces:

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Theorem 3.10 (Botelho, Michels, Pellegrino). (See [11].) Let 1  p  q < ∞ and X1 , . . . , Xn be L∞ -spaces. Then Lm,p (X1 , . . . , Xn ; Y ) ⊂ Lm,q (X1 , . . . , Xn ; Y ). A different behavior, but not less intriguing, occurs with “inclusion results” for the class of absolutely summing multilinear operators. If 1  p < ∞, T ∈ L(X1 , . . . , Xn ; Y ) is absolutely p-summing (T ∈ Las,p (X1 , . . . , Xn ; Y )) when (1) (n)

∞ ∈ p (Y ) T x j , . . . , xj j =1 w for every (xj(k) )∞ j =1 ∈ p (Xk ). The following result shows that in some cases the Inclusion Theorem is exactly the opposite from the linear case (i.e. if p  q implies Las,q ⊂ Las,p ):

Theorem 3.11 (Junek, Matos, Pellegrino). (See [39].) If X has cotype 2, Y is any Banach space and n  2, then

Las,q n X; Y ⊂ Las,p n X; Y holds true for 1  p  q  2. We finish this brief overview with the particular aspects of absolutely summability in the context of multilinear operators with the following variation of the concept multiple summing operators introduced in [13]: If p  1, T ∈ L(X1 , . . . , Xn ; Y ) is strongly multiple p-summing (T ∈ Lsm,p (X1 , . . . , Xn ; Y )) if there exists C  0 such that 

m 

 (1)

 T x , . . . , x (n) p

j1 ,...,jn =1

j1

jn



1/p C

sup

m 

φ∈BL(X1 ,...,Xn ;K) j ,...,j =1 n 1

 (1)

 φ x , . . . , x (n) p j1 jn

1/p

(l)

for every m ∈ N, xjl ∈ Xl with l = 1, . . . , n and jl = 1, . . . , m. Some open problems related to this class are recently presented in [60]. For example: • Is there an Inclusion Theorem for this class? • Does the class Lsm,p coincide with the ideal of strongly (p, p)-summing multilinear operators (see Definition 3.17)? The development of the nonlinear theory of absolutely summing operators leads to the search for nonlinear versions of the Pietsch Domination–Factorization Theorem (see, e.g., [1,15,16, 32,33,47]). Recently, in [17] (see also an addendum in [58] and [59] for a related result), an abstract unified approach to Pietsch-type results was presented as an attempt to show that all the known Pietsch-type theorems were particular cases of a unified general version. However, these approaches were not complete, as we will show later on Section 4.

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3.2. Applications to the theory of absolutely summing multilinear operators The multilinear theory of absolutely summing mappings seems to have its starting point in 1930/1931 with Littlewood’s 4/3 Theorem [45] and the Bohnenblust–Hille Theorem [8]. Precisely, the Bohnenblust–Hille Theorem shows that for every positive integer n the following inequality holds 

N   2n  T (ei , . . . , ei ) n+1 n 1

 n+1 2n

n

n+1 2n

2

n−1 2

T 

(3.4)

i1 ,...,in =1 N for every n-linear mapping T : N ∞ × · · · × ∞ → C and every positive integer N . The case n = 2 is Littlewood’s 4/3 Theorem. Multiple summing multilinear operators have a strong connection with these famous results. For example, in 2003 D. Pérez-García showed that the Bohnenblust– Hille Theorem could be naturally re-written as:

Theorem 3.12 (Bohnenblust–Hille—reformulated). Every continuous n-linear form is multiple 2n ( n+1 , 1)-summing. It is worth mentioning that the Bohnenblust–Hille Theorem was overlooked for a long time and the interest in the multilinear theory related to absolutely summing operators was only recovered in the 1980’s, motivated by A. Pietsch’s work [67]; it is interesting to note that the particular case of multiple summing operators became unexplored until 2003. The first concepts explored in the 80’s were the notions of absolutely summing multilinear operators (see (3.3)) and dominated multilinear operators, which is a particular case of (3.3) for special choices of p, q1 , . . . , qn . Recently basic results and applications have appeared, mainly related to the notion of multiple summability. With respect to intrinsic results of the theory, it must be mentioned that Grothendieck’s Theorem 3.2 is valid for multiple summing operators: Theorem 3.13 (Pérez-García). (See [62].) If 1  p  2, then



Lm,p n 1 ; 2 = L n 1 ; 2 . In [14] a general method of obtaining coincidence situations for multiple summing multilinear operators is presented, including a new proof of Theorem 3.13. Another work that deserves special credit is a recent paper due to A. Defant, D. Popa and U. Schwarting which shows an intriguing extension of the Bohnenblust–Hille Theorem involving cotype; this paper seems to be a definitive reference concerning the Bohnenblust–Hille Theorem. Below, an operator T ∈ L(X1 , . . . , Xn ; Y ) is separately (r, 1)-summing if it is absolutely (r, 1)summing in each coordinate separately: Theorem 3.14 (Defant, Popa, Schwarting—Bohnenblust–Hille generalized). Let Y be a Banach space with cotype q, and 1  r < q. Then each separately (r, 1)-summing T ∈ L(X1 , . . . , Xn ; Y ) qrn is multiple ( q+(n−1)r , 1)-summing. In particular, if Y = K, q = 2 and r = 1 we have the Bohnenblust–Hille Theorem in the context of multiple summing operators.

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265

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Concerning connections and applications of some of the multilinear “prototypes” of absolutely summing operators in other contexts we can mention the following ones: • For every n ∈ N, a tensor norm of order n (see [63]) is shown in [23] to preserve unconditionality for Lp -spaces. • Defant et al. [22] provide optimal estimates for the width of Bohr’s strip for Dirichlet series in infinite-dimensional Banach spaces. • Pérez-García et al. [64] obtain applications to quantum information theory; they prove that, contrary to the bipartite case, tripartite Bell inequalities can be unboundedly violated. • Jarchow et al. [38] prove that the existence of Hahn–Banach-type extension theorems for multilinear forms is connected to structural properties of the underlying spaces. • Acosta et al. [2] relate the notion of multiple summing multilinear operators with the theory of norm attaining multilinear maps. For other interesting approaches related to the multilinear theory of absolutely summing operators we refer to [4,24,61]. For non-multilinear approaches we refer to [10,39,50,51]. This section will actually show that for multilinear mappings there exists an improved version of the Inclusion Principle (we just need to explore the multi-linearity). For technical reasons the present abstract setting is slightly different from the one of the previous section. Let X, Y , V , G, W be (arbitrary) non-void sets, Z a vector space and H ⊂ Map(X, Y ). Consider the arbitrary mappings R : Z × G × W → [0, ∞), S : H × Z × G × V → [0, ∞). Let 1  p  q < ∞ and α ∈ R. Suppose that

sup

m 

R(zj , gj , w)p < ∞

and

sup

w∈W j =1

m 

S(f, zj , gj , v)q < ∞

v∈V j =1

for every positive integer m and z1 , . . . , zm ∈ Z and g1 , . . . , gm ∈ G. We will say that f ∈ H is (q, p)-abstract (R, S)-summing (notation f ∈ RS(q,p) ) if there is a constant C > 0 so that  sup

m 

v∈V j =1

1 q

q

S(f, zj , gj , v)

  C sup

m 

1/p p

R(zj , gj , w)

,

(3.5)

w∈W j =1

for all z1 , . . . , zm ∈ Z, g1 , . . . , gm ∈ G and m ∈ N. We will say that S and R are multiplicative in the variable Z if R(λz, g, w) = |λ|R(z, g, w), S(f, λz, g, v) = |λ|S(f, z, g, v).

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Theorem 3.15. Let pj and qj be as in (3.1) and suppose that S and R are multiplicative in the variable Z. Then RS(q1 ,p1 ) ⊂ RS(q2 ,p2 ) . Proof. If p1 = p2 = p the result is clear. So, let us consider p1 < p2 (and hence q1 < q2 ). If f ∈ RS(q1 ,p1 ) , there is a C > 0 such that  sup

m 

1

q1

q1

S(f, zj , gj , v)

  C sup

v∈V j =1

w∈W

m 

 p1

1 p1

R(zj , gj , w)

(3.6)

,

j =1

for all z1 , . . . , zm ∈ Z, g1 , . . . , gm ∈ G and m ∈ N. Then  sup

m 

1

q1

q1

S(f, λj zj , gj , v)

  C sup

v∈V j =1

w∈W

m 

 p1

1 p1

R(λj zj , gj , w)

,

(3.7)

j =1

for all z1 , . . . , zm ∈ Z, λ1 , . . . , λm ∈ K, g1 , . . . , gm ∈ G and m ∈ N. Define p, q by 1 1 1 = − p p1 p2

1 1 1 = − . q q1 q2

and

So we have 1  q  p < ∞; let m ∈ N, z1 , z2 , . . . , zm ∈ Z and g1 , . . . , gm ∈ G be fixed. For each j = 1, . . . , m, consider λj : V → [0, ∞), q2

λj (v) := S(f, zj , gj , v) q . So, recalling that S is multiplicative in Z, we have 

m 



1 S(f, zj , gj , v)q2

q1

=

j =1

m 

q S f, λj (v)zj , gj , v 1

1

q1

j =1

  C sup w∈W

m 

p R λj (v)zj , gj , w 1



1 p1

j =1

for every v ∈ V . Since R is multiplicative in Z and, as we did before, from Hölder’s Inequality we obtain 

m  j =1

1 S(f, zj , gj , v)q2

q1

  C sup w∈W

m  j =1

 λj (v)p1 R(zj , gj , w)p1

1 p1

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265

  C sup w∈W

 =C

m 

 p1  p

λj (v)p

j =1

m 

m 

1249

 p1  p2

R(zj , gj , w)p2

1 p1

j =1





1 p

λj (v)p

sup w∈W

j =1

m 

 R(zj , gj , w)p2

1 p2

j =1

for every v ∈ V . Since p  q we have .p  .q and then 

m 



1

q1

q2

S(f, zj , gj , v)

C

j =1

m 

1 sup

λj (v)

w∈W

j =1

 =C

m 



q

q

m 

 p2

R(zj , gj , w)

j =1

1



q

q2

sup

S(f, zj , gj , v)

w∈W

j =1

for every v ∈ V and we easily conclude the proof.

1 p2

m 

 p2

1 p2

R(zj , gj , w)

j =1

2

Let us show how the above result applies to the multilinear theory of absolutely summing mappings. Our intention is illustrative rather than exhaustive. For the theory of multilinear mappings between Banach spaces we refer to [29,55]. Consider the following concepts of multilinear summability for 1  p  q < ∞ (inspired in [19,28]): Definition 3.16. A mapping T ∈ L(X1 , . . . , Xn ; Y ) is (q, p)-semi-integral if there exists C  0 such that 

m   1

 T x , . . . , x n q j



1/q

j

C

j =1

sup

m   1  ϕ1 x . . . ϕn x n p

ϕl ∈BX∗ ,l=1,...,n j =1

j

1/p

j

(3.8)

l

for every m ∈ N, xjl ∈ Xl with l = 1, . . . , n and j = 1, . . . , m. In the above situation we write T ∈ Lsi(q,p) (X1 , . . . , Xn ; Y ). Definition 3.17. A mapping T ∈ L(X1 , . . . , Xn ; Y ) is strongly (q, p)-summing if there exists C  0 such that 

m   1

 T x , . . . , x n q j

j =1

j



1/q C

sup

m   1

 ϕ x , . . . , x n p

ϕ∈BL(X1 ,...,Xn ;K) j =1

j

1/p

j

for every m ∈ N, xjl ∈ Xl with l = 1, . . . , n and j = 1, . . . , m. In the above situation we write T ∈ Lss(q,p) (X1 , . . . , Xn ; Y ). For these concepts there are natural Pietsch Domination-type theorems:

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Theorem 3.18. (See [19].) A map T ∈ L(X1 , . . . , Xn ; Y ) is (p, p)-semi-integral if and only if there exist C  0 and a regular probability measure μ on the Borel σ -algebra B(BX1∗ × · · · × BXn∗ ) of BX1∗ × · · · × BXn∗ endowed with the product of the weak star topologies σ (Xl∗ , Xl ), l = 1, . . . , n, such that   T (x1 , . . . , xn )  C



1/p   ϕ1 (x1 ) . . . ϕn (xn )p dμ(ϕ1 , . . . , ϕn )



BX∗ ×···×BX∗ n

1

for every xj ∈ Xj and j = 1, . . . , n. Theorem 3.19 (Dimant). (See [28].) A map T ∈ L(X1 , . . . , Xn ; Y ) is strongly (p, p)-summing if and only if there are a probability measure μ on B(E1 ⊗π ···⊗π En )∗ , with the weak-star topology, and a constant C  0 so that   T (x1 , . . . , xn )  C





1 p   ϕ(x1 ⊗ · · · ⊗ xn )p dμ(ϕ)

(3.9)

B(X1 ⊗π ···⊗π Xn )∗

for all (x1 , . . . , xn ) ∈ X1 × · · · × Xn . As corollaries, the following inclusion results hold: Proposition 3.20. If 1  p  q < ∞, then, for any Banach spaces X1 , . . . , Xn , Y , the following inclusions hold: Lsi(p,p) (X1 , . . . , Xn ; Y ) ⊂ Lsi(q,q) (X1 , . . . , Xn ; Y ),

and

Lss(p,p) (X1 , . . . , Xn ; Y ) ⊂ Lss(q,q) (X1 , . . . , Xn ; Y ). However, the Pietsch Domination Theorem is of no use for other choices of pj , qj . But, as it will be shown, in this case the multilinearity allows us to obtain better results than those from Theorem 2.1. For the class of semi-integral mappings we may choose Z = X1 , G = X2 × · · · × Xn , W = BX1∗ × · · · × BXn∗ , V = {0}, H = L(X1 , . . . , Xn ; Y ) and consider the mappings R : Z × G × W → [0, ∞), 

 R x1 , (x2 , . . . , xn ), (ϕ1 , . . . , ϕn ) = ϕ1 (x1 ) . . . ϕn (xn ) and S : H × Z × G × V → [0, ∞), 

 S T , x1 , (x2 , . . . , xn ), 0 = T (x1 , . . . , xn ). The case of the class of strongly summing multilinear mappings is analogous. So, as a consequence of Theorem 3.15, we have:

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Proposition 3.21. If pj and qj are as in (3.1) then, for any Banach spaces X1 , . . . , Xn , Y , the following inclusions hold: Lsi(q1 ,p1 ) (X1 , . . . , Xn ; Y ) ⊂ Lsi(q2 ,p2 ) (X1 , . . . , Xn ; Y ),

and

Lss(q1 ,p1 ) (X1 , . . . , Xn ; Y ) ⊂ Lss(q2 ,p2 ) (X1 , . . . , Xn ; Y ). 3.3. Applications to non-multilinear absolutely summing operators As in the previous section, we intend to illustrate how the Inclusion Principle can be invoked in other situations; we have no exhaustive purpose. Let us consider the following definitions extending the notion of semi-integral and strongly multilinear mappings to the non-multilinear context, even with spaces having a less rich structure than a Banach space: Definition 3.22. Let X1 , . . . , Xn be normed spaces and Y = (Y, d) be a metric space. An arbitrary map f : X1 × · · · × Xn → Y is ((q, α), p)-semi-integral at (a1 , . . . , an ) ∈ X1 × · · · × Xn (notation f ∈ Mapsi((q,α),p) (X1 , . . . , Xn ; Y )) if there exists C  0 such that 

m 

q d f a1 + xj1 , . . . , an + xjn , f (a1 , . . . , an )

1/α

j =1

C

sup

m   1  ϕ1 x . . . ϕn x n p

ϕl ∈BX∗ ,l=1,...,n j =1

j

j

l

for every m ∈ N, xjl ∈ Xl with l = 1, . . . , n and j = 1, . . . , m. Definition 3.23. Let X1 , . . . , Xn be normed spaces and Y = (Y, d) be a metric space. An arbitrary map f : X1 × · · · × Xn → Y is strongly ((q, α), p)-summing at (a1 , . . . , an ) ∈ X1 × · · · × Xn (notation f ∈ Mapss((q,α),p) (X1 , . . . , Xn ; Y )) if there exists C  0 such that 

m 

q d f a1 + xj1 , . . . , an + xjn , f (a1 , . . . , an )

1/α

j =1

C

sup

m   1

 ϕ x , . . . , x n p j j

ϕ∈L(X1 ,...,Xn ;K) j =1

for every m ∈ N, xjl ∈ Xl with l = 1, . . . , n and j = 1, . . . , m. By choosing adequate parameters in Theorem 2.1 we obtain: Theorem 3.24. If pj and qj satisfy (3.1), then Mapsi((q1 ,1),p1 ) (X1 , . . . , Xn ; Y ) ⊂ Mapsi((q2 ,α),p2 ) (X1 , . . . , Xn ; Y ), Mapss((q1 ,1),p1 ) (X1 , . . . , Xn ; Y ) ⊂ Mapss((q2 ,α),p2 ) (X1 , . . . , Xn ; Y )

and

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for α=

q 2 p1 q 1 p2

if p1 < p2 .

3.4. Applications to non-multilinear absolutely summing operators in the sense of Matos In [50] M. Matos considered a concept of summability which can be characterized by means of an inequality as follows: If X and Y are Banach spaces, a map f : X → Y is absolutely (q, p)-summing at a if there are constants C > 0 and δ > 0 such that ∞ ∞       f (a + zj ) − f (a)q  C sup ϕ(zj )p , ϕ∈BX∗ j =1

j =1 u for all (zj )∞ j =1 ∈ p (X) and

  (zj )∞ 



j =1 w,p

:= sup

ϕ∈BX∗

∞    ϕ(zj )p

1/p < δ.

j =1

Above,     weak  up (X) := (zj )∞ (X); lim (zj )∞ j =n w,p = 0 . j =1 ∈ p n→∞

It is worth mentioning that there exists a version of our Inclusion Principle in this context. If α ∈ R, we will say that f : X → Y is Matos absolutely ((q, α), p)-summing at a (denoted by f ∈ M((q,α),p) ) if there are constants C > 0 and δ > 0 such that 

∞    f (a + zj ) − f (a)q

1

α

 C sup

∞    ϕ(zj )p ,

(3.10)

ϕ∈BX∗ j =1

j =1

∞ u for all (zj )∞ j =1 ∈ p (X) and (zj )j =1 w,p < δ. If α = 1 we recover Matos’ original concept and simply write (q, p) instead of ((q, 1), p). With this at hand, we can now state the following result:

Theorem 3.25. If pj and qj are as in (3.1), then M(q1 ,p1 ) ⊂ M((q2 ,α),p2 ) for α= whenever p1 < p2 .

q 2 p1 q 1 p2

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265

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4. A full general version of the Pietsch Domination Theorem From now on all measures considered in this paper will be probability measures defined in the Borel sigma-algebras of compact topological spaces. In this section, and for the sake of completeness, we will recall the more general version that we know, until now, for the Pietsch Domination Theorem. This approach is a combination of [17] and a recent improvement from [58], and will be generalized in the subsequent section. Let X, Y and E be (arbitrary) non-void sets, H be a family of mappings from X to Y , G be a Banach space and K be a compact Hausdorff topological space. Let R : K × E × G → [0, ∞)

and S : H × E × G → [0, ∞)

be mappings so that the following property hold: “The mapping Rx,b : K → [0, ∞)

defined by Rx,b (ϕ) = R(ϕ, x, b)

is continuous for every x ∈ E and b ∈ G.” Let R and S be as above and 0 < p < ∞. A mapping f ∈ H is said to be R–S-abstract p-summing if there is a constant C > 0 so that 

m 

1



p

S(f, xj , bj )p

 C sup ϕ∈K

j =1

m 

1

p

R(ϕ, xj , bj )p

,

(4.1)

j =1

for all x1 , . . . , xm ∈ E, b1 , . . . , bm ∈ G and m ∈ N. The general unified PDT reads as follows: Theorem 4.1. (See [17,58].) Let R and S be as above, 0 < p < ∞ and f ∈ H. Then f is R–Sabstract p-summing if and only if there is a constant C > 0 and a Borel probability measure μ on K such that  S(f, x, b)  C

1 p

R(ϕ, x, b) dμ

p

(4.2)

K

for all x ∈ E and b ∈ G. From now on, if X1 , . . . , Xn , Y are arbitrary sets, Map(X1 , . . . , Xn ; Y ) will denote the set of all arbitrary mappings from X1 × · · · × Xn to Y (no assumption is necessary). Let 0 < q1 , . . . , qn < ∞ and 1/q = nj=1 1/qj . A map f ∈ Map(X1 , . . . , Xn ; Y ) is (q1 , . . . , qn )-dominated at (a1 , . . . , an ) ∈ X1 × · · · × Xn if there is a C > 0 and there are Borel probabilities μk on BXk∗ , k = 1, . . . , n, such that

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D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265 n   

f a1 + x (1) , . . . , an + x (n) − f (a1 , . . . , an )  C



 (k) qk ϕ x  dμk

1

qk

(4.3)

k=1 B ∗ X k

for all x (j ) ∈ Xj , j = 1, . . . , n. In our recent note [59] we observed that the general approach from [17,58] was not able to characterize the mappings satisfying (4.3), and a new Pietsch-type theorem was proved: Theorem 4.2. (See [59].) A map f ∈ Map(X1 , . . . , Xn ; Y ) is (q1 , . . . , qn )-dominated at (a1 , . . . , an ) ∈ X1 × · · · × Xn if and only if there is a C > 0 such that 

m     (1)

b . . . b(n) f a1 + x (1) , . . . , an + x (n) − f (a1 , . . . , an ) q j

j

j =1

C

n 

j

 sup

k=1 ϕ∈BXk∗

j

m   (k)  (k)  qk b ϕ x  j

1/q

1/qk

j

(4.4)

j =1 (k)

(k)

for every positive integer m, (xj , bj ) ∈ Xk × K, with (j, k) ∈ {1, . . . , m} × {1, . . . , n}. (k)

As pointed in [59], inequality (4.4) arises the curious idea of weighted summability: each xj (k)

is interpreted as having a “weight” bj and in this context the respective sum  

f a1 + x (1) , . . . , an + x (n) − f (a1 , . . . , an ) j

(1)

j

(n)

inherits a weight |bj · · · · · bj |. As it is shown in [17], the unified PDT (UPDT) immediately recovers several known Pietschtype theorems. However, in at least one important situation (the PDT for dominated multilinear mappings), the respective PDT is not straightforwardly obtained from the UPDT from [17]. In fact, as pointed in [59], the structural difference between (4.2) and (4.3) is an obstacle to recover some domination theorems as Theorem 4.2. The same deficiency of the (general) UPDT will be clear in Section 4.4. In the next section the approach of [59] is translated to a more abstract setting and the final result shows that Theorem 4.2 holds in a very general context. Some applications are given in order to show the reach of this generalization. 4.1. The full general Pietsch Domination Theorem In this section we prove a quite general PDT which seems to delimit the possibilities of such kind of result. The procedure is an abstraction of the main result of [59]. It is curious the fact that the Unified Pietsch Domination Theorem from [17] does not use Pietsch’s original argument, but this more general version, as in [59], uses precisely Pietsch’s original approach in an abstract disguise. The main tool of our argument, as in Pietsch’s original proof of the linear case, is a lemma by Ky Fan.

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265

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Lemma 4.3 (Ky Fan). Let K be a compact Hausdorff topological space and F be a concave family of functions f : K → R which are convex and lower semi-continuous. If for each f ∈ F there is a xf ∈ K so that f (xf )  0, then there is a x0 ∈ K such that f (x0 )  0 for every f ∈ F . Let X1 , . . . , Xn , Y and E1 , . . . , Er be (arbitrary) non-void sets, H be a family of mappings from X1 × · · · × Xn to Y . Let also K1 , . . . , Kt be compact Hausdorff topological spaces, G1 , . . . , Gt be Banach spaces and suppose that the maps 

Rj : Kj × E1 × · · · × Er × Gj → [0, ∞), j = 1, . . . , t, S : H × E1 × · · · × Er × G1 × · · · × Gt → [0, ∞)

satisfy: (1) For each x (l) ∈ El and b ∈ Gj , with (j, l) ∈ {1, . . . , t} × {1, . . . , r} the mapping

defined by (Rj )x (1) ,...,x (r) ,b (ϕ) = Rj ϕ, x (1) , . . . , x (r) , b

(Rj )x (1) ,...,x (r) ,b : Kj → [0, ∞) is continuous. (2) The following inequalities hold: 





Rj ϕ, x (1) , . . . , x (r) , ηj b(j )  ηj Rj ϕ, x (1) , . . . , x (r) , b(j ) ,

(4.5) S f, x (1) , . . . , x (r) , α1 b(1) , . . . , αt b(t)  α1 . . . αt S f, x (1) , . . . , x (r) , b(1) , . . . , b(t)

for every ϕ ∈ Kj , x (l) ∈ El (with l = 1, . . . , r), 0  ηj , αj  1, bj ∈ Gj , with j = 1, . . . , t and f ∈ H. Definition 4.4. If 0 < p1 , . . . , pt , p < ∞, with p1 = p11 + · · · + p1t , a mapping f : X1 × · · · × Xn → Y in H is said to be R1 , . . . , Rt –S-abstract (p1 , . . . , pt )-summing if there is a constant C > 0 so that 

m  (1) (r) (1) (t) p S f, xj , . . . , xj , bj , . . . , bj

1

p

j =1

C

t 

 sup

k=1 ϕ∈Kk (s)

(s)

(l)

m 

(1) (r) (k) pk Rk ϕ, xj , . . . , xj , bj



1 pk

(4.6)

j =1 (l)

for all x1 , . . . , xm ∈ Es , b1 , . . . , bm ∈ Gl , m ∈ N and (s, l) ∈ {1, . . . , r} × {1, . . . , t}. The proof mimics the steps of the particular case proved in [59], and hence we omit some details. Due the more abstract environment, the new proof has extra technicalities but just in the final part of the proof a more important care will be needed when dealing with the parameter β.

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As in the proof of [59], we need the following lemma (see [36, p. 17]): Lemma 4.5. Let 0 < p1 , . . . , pn , p < ∞ be so that 1/p =

n

j =1 1/pj .

Then

n n 1  p  1 pj qj  q p pj j j =1

j =1

regardless of the choices of q1 , . . . , qn  0. Now we are ready to prove the aforementioned theorem: Theorem 4.6. A map f ∈ H is R1 , . . . , Rt –S-abstract (p1 , . . . , pt )-summing if and only if there is a constant C > 0 and Borel probability measures μj on Kj such that t 

S f, x (1) , . . . , x (r) , b(1) , . . . , b(t)  C





p Rj ϕ, x (1) , . . . , x (r) , b(j ) j dμj

1/pj (4.7)

j =1 K j

for all x (l) ∈ El , l = 1, . . . , r and b(j ) ∈ Gj , with j = 1, . . . , t. Proof. One direction is canonical and we omit. Let us suppose that f ∈ H is R1 , . . . , Rt – S-abstract (p1 , . . . , pt )-summing. Consider the compact sets P (Kk ) of the probability mea(l) (s) m sures in C(Kk )∗ , for all k = 1, . . . , t. For each (xj )m j =1 in El and (bj )j =1 in Gs , with (s, l) ∈ {1, . . . , t} × {1, . . . , r}, let g = g(x (l) )m j

(s) m j =1 ,(bj )j =1 ,(s,l)∈{1,...,t}×{1,...,r}

: P (K1 ) × · · · × P (Kt ) → R

be defined by

m

 1 (1) (r) (1) (t) p t S f, xj , . . . , xj , bj , . . . , bj g (μj )j =1 = p j =1

  t  1 (1) (r) (k) pk −C Rk ϕ, xj , . . . , xj , bj dμk . pk p

k=1

Kk

As usual, the family F of all such g’s is concave and one can also easily prove that every g ∈ F g is convex and continuous. Besides, for each g ∈ F there are measures μj ∈ P (Kj ), j = 1, . . . , t, such that g g g μ1 , . . . , μt  0. In fact, using the compactness of each Kk (k = 1, . . . , t), the continuity of (Rk )x (1) ,...,x (r) ,b(k) , there are ϕk ∈ Kk so that

j

j

j

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265 m 

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m  (1) (r) (k) pk (1) (r) (k) pk Rk ϕk , xj , . . . , xj , bj = sup Rk ϕ, xj , . . . , xj , bj . ϕ∈Kk j =1

j =1 g

Now, with the Dirac measures μk = δϕk , k = 1, . . . , t, and Lemma 4.5 we get g g g μ1 , . . . , μt  0. So, Ky Fan’s Lemma asserts that there are μj ∈ P (Kj ), j = 1, . . . , t, so that g(μ1 , . . . , μt )  0 for all g ∈ F . Hence  m   1 (1) (r) (1) (t) p S f, xj , . . . , xj , bj , . . . , bj p j =1

  m t 

p 1 −C Rk ϕ, xj(1) , . . . , xj(r) bj(k) k dμk  0 pk p

Kk j =1

k=1

and from the particular case m = 1 we obtain t



k=1

Kk

 1

p 1 S f, x (1) , . . . , x (r) , b(1) , . . . , b(t)  C p p pk



p Rk ϕ, x (1) , . . . , x (r) , b(k) k dμk .

(4.8)

If x (1) , . . . , x (r) , b(1) , . . . , b(t) are given and, for k = 1, . . . , t, define 



p Rk ϕ, x (1) , . . . , x (r) , b(k) k dμk

τk :=

1/pk .

Kk

If τk = 0 for every k then, the result is immediate. Let us now suppose that τj is not zero for some j ∈ {1, . . . , t}. Consider   V = j ∈ {1, . . . , t}; τj = 0 and β > 0 big enough to get 1 −1 0 < τj β ppj < 1 for every j ∈ V .

The above condition is necessary in view of (4.5). Consider, also,  ϑj =

1

(τj β ppj )−1 1

if j ∈ V , if j ∈ / V.

(4.9)

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Thus, since 0 < ϑj  1, we have t



k=1

Kk

 1

p 1 S f, x (1) , . . . , x (r) , ϑ1 b(1) , . . . , ϑt b(t)  C p p pk



p Rk ϕ, x (1) , . . . , x (r) , ϑk b(k) k dμk

 1 1 −pk pk τk β ppk  Cp τk pk k∈V



Cp

1 p β p1

and   p

p S f, x (1) , . . . , x (r) , b(1) , . . . , b(t)  C p β ( j ∈V 1/pj )−1/p τj .

(4.10)

j ∈V

If V = {1, . . . , t}, then 1  1 > 0. − p pj j ∈V

Note that it is possible to make β → ∞ in (4.10), since it does not contradict (4.9); so we get

p S f, x (1) , . . . , x (r) , b(1) , . . . , b(t) = 0 and we again reach (4.7). The case V = {1, . . . , t} is immediate.

2

4.2. Application: the (general) unified PDT and the case of dominated multilinear mappings By choosing r = t = n = 1 in Theorem 4.6 we obtain an improvement of the Unified Pietsch Domination Theorem from [17]. In fact, we obtain precisely [58, Theorem 2.1] which is essentially the general unified PDT (we just need to repeat the trick used in [58, Theorem 3.1]). Thus, we obtain Theorem 4.1 (Section 4): Theorem 4.7. (See [58].) Let R and S be as in Section 4, 0 < p < ∞ and f ∈ H. Then f is R– S-abstract p-summing if and only if there is a constant C > 0 and a Borel probability measure μ on K such that  S(f, x, b)  C K

for all x ∈ E and b ∈ G.

1 R(ϕ, x, b)p dμ

p

,

(4.11)

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It is interesting to notice that, for n > 1 and with the trick used in [58, Theorem 3.1], essentially what emerges is the notion of weighted summability. In resume, this trick works perfectly for n = 1, but for other cases it forces us to deal with weighted summability. So, one shall not expect for the possible relaxation of conditions (4.5) for the validity of Theorem 4.6. As pointed out in the introduction, and in opposition to what happens in [17], our theorem recovers straightforwardly the domination theorem for (q1 , . . . , qn )-dominated n-linear mappings (with 1/q = 1/q1 + · · · + 1/qn ). In fact, we just need to choose ⎧ t = n, ⎪ ⎪ ⎪ ⎪ Gj = Xj and Kj = BX∗ for all j = 1, . . . , n, ⎪ j ⎪ ⎪ ⎪ ⎪ E = K, j = 1, . . . , r, j ⎪ ⎨ H = L(X1 , . . . , Xn ; Y ), ⎪ ⎪ pj = qj for all j = 1, . . . , n, ⎪ ⎪ ⎪



 ⎪ ⎪ S T , x (1) , . . . , x (r) , b(1) , . . . , b(n) = T b(1) , . . . , b(n) , ⎪ ⎪ ⎪



 ⎩ Rk ϕ, x (1) , . . . , x (r) , b(k) = ϕ b(k)  for all k = 1, . . . , n. So, with these choices, T is R1 , . . . , Rn –S-abstract (q1 , . . . , qn )-summing precisely when T is (q1 , . . . , qn )-dominated. In this case Theorem 4.6 tells us that there is a constant C > 0 and there are measures μk on Kk , k = 1, . . . , n, so that n 

(1) (r) (1) (n) S T,x ,...,x ,b ,...,b C





q Rk ϕ, x (1) , . . . , x (r) , b(k) k dμk

1

qk

,

k=1 K k

i.e., n   (1)

 T b , . . . , b(n)   C



 (k) qk ϕ b  dμk

1

qk

.

k=1 K k

4.3. Application: the Farmer–Johnson Domination Theorem In [17] it is sketched how the Farmer–Johnson Domination Theorem [32] can be recovered from the unified PDT obtained in Section 4.2. For the sake of completeness we will show, with some details, how this can be done through an easier procedure. If M = (M, dM ) and N = (N, dN ) are metric spaces, a map T : M → N is Lipschitz psumming (Farmer and Johnson [32]) if there is a constant C such that, for all natural n, positive real numbers a1 , . . . , an and x1 , . . . , xn , y1 , . . . , yn ∈ M, n  i=1

n  p 

p ai dN T (xi ), T (yi )  C sup ai f (xi ) − f (yi ) , f ∈BM # i=1

where BM # is the unit ball of the Lipschitz dual M # of M. Note that T is Lipschitz p-summing if and only if it is R–S-abstract p-summing with

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E = M × M, G = R, and K = BM # , which is a compact Hausdorff space in the topology of pointwise convergence on M, H is the set of all mappings from M to N and R and S are given by:  

R f, (x, y), a = |a|1/p f (x) − f (y),



S T , (x, y), a = |a|1/p dN T (x), T (y) .

R : BM # × (M × M) × R → [0, ∞), S : H × (M × M) × R → [0, ∞),

Thus, we have, as a consequence of Theorem 4.7: Theorem 4.8 (Farmer–Johnson). The following are equivalent for a mapping T : M → N between metric spaces: (i) T is Lipschitz p-summing. (ii) There is a probability μ on BM # and a constant C  0 such that   dN (T x, T y)  C

1/p   f (x) − f (y)p dμ(f )

BM #

for all x, y ∈ M. 4.4. Application: the PDT for Cohen strongly q-summing operators The class of Cohen strongly q-summing multilinear operators was introduced by D. Achour and L. Mezrag in [1]. Let 1 < q < ∞ and X1 , . . . , Xn , Y arbitrary Banach spaces. If q > 1, then q ∗ denotes the real number satisfying 1/q + 1/q ∗ = 1. A continuous n-linear operator T : X1 × · · · × Xn → Y is Cohen strongly q-summing if and only if there is a constant C > 0 (j ) (j ) ∗ in Y ∗ , such that for any positive integer m, x1 , . . . , xm in Xj (j = 1, . . . , n) and any y1∗ , . . . , ym the following inequality hold:  m n 1/q m     (j ) q  ∗ (1)

 (n) C y T x , . . . , x x  i i i i i=1

i=1 j =1

 sup

y ∗∗ ∈BY ∗∗

m   ∗∗ ∗ q ∗ y y  i

1/q ∗ .

i=1

In the same paper the authors also prove the following Pietsch-type theorem: Theorem 4.9 (Achour–Mezrag). A continuous n-linear mapping T : X1 ×· · ·×Xn → Y is Cohen strongly q-summing if and only if there is a constant C > 0 and a probability measure μ on BY ∗∗ so that for all (x (1) , . . . , x (n) , y ∗ ) in X1 × · · · × Xn × Y ∗ the inequality  n   1∗   ∗ (1)   ∗∗ ∗ q ∗ q

 y T x , . . . , x (n)   C x (k)  y y  dμ k=1

is valid.

BY ∗∗

(4.12)

D. Pellegrino et al. / Advances in Mathematics 229 (2012) 1235–1265

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Note that by choosing the parameters ⎧ t = 2 and r = n − 1, ⎪ ⎪ ⎪ ⎪ Ei = Xi for all i = 1, . . . , n − 1, ⎪ ⎪ ⎪ ⎪ ⎪ K 1 = BX1∗ ×···×Xn∗ and K2 = BY ∗∗ , ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎨ G1 = Xn and G2 = Y , H = L(X1 , . . . , Xn ; Y ), ⎪ ⎪ p = 1, p1 = q and p2 = q ∗ , ⎪ ⎪

 ∗ (1)

 ⎪ ⎪ (1) (n) ∗ ⎪ S T , x = y T x , . . . , x (n) , , . . . , x , y ⎪ ⎪     ⎪

⎪ ⎪ ⎪ R1 ϕ, x (1) , . . . , x (n) = x (1)  · · · x (n) , ⎪ ⎪

  ⎩ R2 ϕ, x (1) , . . . , x (n−1) , y ∗ = ϕ y ∗ , we can easily conclude that T : X1 × · · · × Xn → Y is Cohen strongly q-summing if and only if T is R1 , R2 –S-abstract (q, q ∗ )-summing. Theorem 4.6 tells us that T is R1 , R2 –S-abstract (q, q ∗ )-summing if and only if there is a C > 0 and there are probability measures μk in Kk , k = 1, 2, such that

S T , x (1) , . . . , x (n) , y ∗ 1  1∗  q q



∗ (1) (n) q (1) (n−1) ∗ q C R1 ϕ, x , . . . , x dμ1 R2 ϕ, x , . . . , x ,y dμ2 , K1

K2

i.e.,  ∗ (1)

 y T x , . . . , x (n)   C





 (1)   (n)  q x  · · · x  dμ1

1   q

BX∗ ×···×X∗ 1

 ∗ q ∗ ϕ y  dμ2



1 q∗

BY ∗∗

n

 

    = C x (1)  · · · x (n) 

 ∗ q ∗ ϕ y  dμ2



1 q∗

BY ∗∗

and we recover (4.12) regardless of the choice of the positive integer m and x (k) ∈ Xk , k = 1, . . . , n. 5. Weighted summability The notion of weighted summability (see the comments right after Theorem 4.2) emerged from [59] as a natural concept when we were dealing with problem (4.3). In this section we observe that this concept in fact emerges in more abstract situations and seems to be unavoidable in further developments of the nonlinear theory.  Let 0 < q1 , . . . , qn < ∞, 1/q = nj=1 1/qj , X1 , . . . , Xn be Banach spaces and A : Map(X1 , . . . , Xn ; Y ) × X1 × · · · × Xn → [0, ∞)

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be an arbitrary map. Let us say that f ∈ Map(X1 , . . . , Xn ; Y ) is A–(q1 , . . . , qn )-dominated if there are a constant C > 0 and measures μj in BXj∗ (with the weak star topology), j = 1, . . . , n, such that

A f, x (1) , . . . , x (n)  C



 (1) q1 ϕ x  dμ1

1

q1



 (n) qn ϕ x  dμk

· ··· ·

BX∗



1 qn

(5.1)

,

BX∗ n

1

regardless of the choice of the positive integer m and x (k) ∈ Xk , k = 1, . . . , n. In fact, more abstract maps could be used in the right-hand side of (5.1). However, since our intention is illustrative rather than exhaustive, we prefer to deal with this more simple case. Theorem 5.1. An arbitrary map f ∈ Map(X1 , . . . , Xn ; Y ) is A–(q1 , . . . , qn )-dominated if and only if there exists C > 0 such that 

m    (1)

b . . . b(n) A f, x (1) , . . . , x (n) q j

j

j

j

1 q

C

j =1

n 

 sup

k=1 ϕ∈BXk∗

m   (k)  (k)  qk b ϕ x  j j

1/qk (5.2)

j =1

for every positive integer m, (xj(k) , bj(k) ) ∈ Xk × K, with (j, k) ∈ {1, . . . , m} × {1, . . . , n}. Proof. Choosing the parameters ⎧ r = t = n, ⎪ ⎪ ⎪ ⎪ Ej = Xj and Gj = K for all j = 1, . . . , n, ⎪ ⎪ ⎪ ⎪ K = BXj∗ for all j = 1, . . . , n, ⎪ ⎪ ⎨ j H = Map(X1 , . . . , Xn ; Y ), ⎪ ⎪ p = q and pj = qj for all j = 1, . . . , n, ⎪ ⎪ ⎪ 



⎪ ⎪ S f, x (1) , . . . , x (n) , b(1) , . . . , b(n) = b(1) . . . b(n) A f, x (1) , . . . , x (n) , ⎪ ⎪ ⎪ 



 ⎩ Rk ϕ, x (1) , . . . , x (n) , b(k) = b(k) ϕ x (k)  for all k = 1, . . . , n, we easily conclude that (5.2) holds if and only if f is R1 , . . . , Rn –S-abstract (q1 , . . . , qn )summing. In this case Theorem 4.6 tells us that there is a constant C > 0 and there are measures μk on Kk , k = 1, . . . , n, such that n 

S T , x (1) , . . . , x (n) , b(1) , . . . , b(n)  C





q Rk ϕ, x (1) , . . . , x (n) , b(k) k dμk

k=1 K k

i.e., n    (1)

b . . . b(n) A f, x (1) , . . . , x (n)  C



 (k)  (k)  qk b ϕ x  dμk

k=1 B ∗ X k

for all (x (k) , b(k) ) ∈ Xk × K, k = 1, . . . , n, and we readily obtain (5.1).

2

1

qk

,

1

qk

,

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Remark 5.2. As we have mentioned before, the procedure of this last section is illustrative. The interested reader can easily find a characterization similar to Theorem 5.1 in the full abstract context of Definition 4.4. 6. Final comments and directions for further research As we mentioned earlier, we believe that the Inclusion Principle can be useful in different contexts of Mathematical Analysis. Besides, since our applications (even applications related to absolutely summing maps) have no exhaustive purpose, this technique may be useful in forthcoming investigations focused on “new nonlinear prototypes” of absolutely summing operators. Concerning the “full Pietsch Domination Theorem” our perception is that this is a potentially definitive result, which can be regarded as a reference of how far one can go with Pietsch Domination-type arguments. This general approach also avoids the multiplication and appearance of apparently different proofs of Pietsch Domination-type theorems (in different contexts) which are, in fact, all consequence of a unique result. The notion of weighted summability is perhaps the concept that can be explored in more directions. So far we have observed that weighted summability is the natural concept in order to have a Pietsch Domination Theorem in “several variables”. However, since this is a very recent concept, there is absolutely no separate investigation of this notion. Connections with other concepts, coincidence results, the effect of cotype of the Banach spaces involved (for example) have never been investigated in the context of weighted summability and might be a fruitful topic for further research.

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