Abstract
The multilinear Hardy-Littlewood inequalities provide estimates for the sum of the coefficients of multilinear forms when . In this paper we investigate the critical and super-critical cases; i.e., when
Key words
Multilinear forms; sequence spaces; inequalities; estimates
Introduction
Littlewood’s theorem assures that for or , we have
for all positive integers and all bilinear forms , where as usual
and denotes with the norm; the exponent cannot be improved (i.e., cannot be replaced by a smaller one). Under an anisotropic viewpoint, the result can be generalized as follows (see Theorem 5.1 in Pellegrino et al. 2017PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ DM & TEIXEIRA EV. 2017. A regularity principle in sequence spaces and applications. Bull Sci Math 141: 802-837.): the inequality
holds for all whenever satisfy
Moreover, if satisfy
then (1) is not possible, i.e., if
then the constant must depend on .From now on, unless stated otherwise, the exponents involved in the inequalities are positive and can be even infinity (in this case the corresponding sum is replaced by the supremum). We also consider The Hardy-Littlewood inequalities for bilinear forms were conceived in 1934 by Hardy and Littlewood (see Theorem 5 in Hardy & Littlewood 1934HARDY G & LITTLEWOOD JE. 1934. Bilinear forms bounded in space [p,q]. Quart J Math 5: 241-254.), as a natural generalization of Littlewood’s inequality. The results of the seminal paper of Hardy and Littlewood, in a modern and somewhat more general presentation, can be summarized by the following two theorems:
Theorem 1. (see Osikiewicz & Tonge 2001OSIKIEWICZ B & TONGE A. 2001. An interpolation approach to Hardy-Littlewood inequalities for norms of operators on sequence spaces. Linear Algebra Appl 331: 1-9. and Aron et al. 2017ARON R, NÚÑEZ-ALARCÓN D, PELLEGRINO D & SERRANO-RODRÍGUEZ D. 2017. Optimal exponents for Hardy-Littlewood inequalities for m-linear operators. Linear Algebra Appl 531: 399-422.) Let , with . The following assertions are equivalent:
-
There is a constant (not depending on ) such that
for all bilinear forms and all positive integers . -
The exponents satisfy
Moreover, the optimal constant is
Theorem 2. (see Pellegrino et al. 2017PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ DM & TEIXEIRA EV. 2017. A regularity principle in sequence spaces and applications. Bull Sci Math 141: 802-837.) Let , with . The following assertions are equivalent:
-
There is a constant (not depending on ) such that
for all bilinear forms and all positive integers . -
The exponents satisfy
and
Since (2) is trivially verified under the conditions of Theorem 1 , we can unify the two theorems as follows:
Theorem 3. Let and , with . The following assertions are equivalent:
-
There is a constant (not depending on ) such that
for all bilinear forms and all positive integers . -
The exponents satisfy
and
In 1981, Praciano-Pereira (see Praciano-Pereira 1981PRACIANO-PEREIRA T. 1981. On bounded multilinear forms on a class of ℓp spaces. J Math Anal Appl 81: 561-568.) extended the Hardy-Littlewood inequalities to -linear forms as follows: if and
there exists a constant (not depending on ) such that for all -linear forms and for all positive integers .When
Dimant and Sevilla-Peris (see Dimant & Sevilla-Peris 2016DIMANT V & SEVILLA-PERIS P. 2016. Summation of coefficients of polynomials on ℓp spaces. Publ Mat 60: 289-310. and Cavalcante 2018CAVALCANTE W. 2018. Some applications of the regularity principle in sequence spaces. Positivity 22: 191-198.) have proved that there exists a constant (not depending on ) such that for all -linear forms and for all positive integers .Both in (3) and (4 the exponents are sharp, i.e., they cannot be replaced by smaller exponents keeping the constant not depending on (this terminology will be used throughout the paper). However, there still remains the question: what about anisotropic versions of (3) and (4) , i.e., variants with eventually different exponents associated to each index? Throughout this paper we shall address this question and related problems.
In Albuquerque et al. 2014ALBUQUERQUE N, BAYART F, PELLEGRINO D & SEOANE-SEPÚLVEDA JB. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740., the anisotropic version of the result of Praciano-Pereira was finally settled (see also Santos & Velanga 2017SANTOS J & VELANGA T. 2017. On the Bohnenblust-Hille inequality for multilinear forms. Results Math 72: 239-244. for a more complete version for the case ):
Theorem 4. (see Theorem 1.2 in Albuquerque et al. 2014ALBUQUERQUE N, BAYART F, PELLEGRINO D & SEOANE-SEPÚLVEDA JB. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740. and Theorem 5.2 in Pellegrino et al. 2017) Let be such that
and The following assertions are equivalent:-
There is a constant (not depending on ) such that
for all -linear forms and all positive integers . -
The inequality
is verified.
The anisotropic version of (4) is still not completely solved, but in Aron et al. 2017ARON R, NÚÑEZ-ALARCÓN D, PELLEGRINO D & SERRANO-RODRÍGUEZ D. 2017. Optimal exponents for Hardy-Littlewood inequalities for m-linear operators. Linear Algebra Appl 531: 399-422. the following partial answer (that also generalizes Theorem 1 ) was obtained:
Theorem 5. (see Theorem 3.2 in Aron et al. 2017) Let and , with
The following assertions are equivalent:-
There is a constant (not depending on ) such that
for all -linear forms and all positive integers -
The exponents satisfy
with
The attentive reader may wonder why the case
is not investigated in the previous results? The reason is simple, because in this case it is easy to prove that if there exists (not depending on ) such that
for all -linear forms and all positive integers , then (i.e., we are forced to deal with the norm, and the result becomes trivial). However, under the anisotropic viewpoint, as a matter of fact, there is no reason to avoid the case (5) and it constitutes a vast field yet to be explored. The first step in this direction is the following:
Theorem 6. (see Theorem 1 in Paulino 2019PAULINO D. 2019. Critical Hardy-Littlewood inequality for multilinear forms. Rend. Circ. Mat. Palermo, II. 69 (2020), 369-380.) For all we have
for all -linear forms and all positive integers , with
for all Moreover, and are sharp and, for the optimal exponents satisfying (7) fulfillThe case considered in Theorem 6 is called critical because it is a special case of (5), and from now on we shall call case (5) super-critical, which is the topic of the present paper. In the next sections we provide a partial solution to the super-critical case for -linear forms and we investigate what are the conditions needed to obtain -linear Hardy-Littlewood inequalities in the super-critical case.
The 3-linear case
We begin this section by presenting two simple, albeit very useful, lemmas that will be used all along the paper.
Two multi-purpose lemmas
For , we define
and by we shall mean . If and , we define The lemmas read as follows:Lemma 7. Let and . If there is a constant (not depending on ) such that
for all -linear forms and all positive integers , then for all -linear forms and all positive integers .Proof. To simplify the notation, we can suppose .
Let suppose that there is a constant such that
for all -linear forms .Given an -linear form , we define the -linear form given by
It is obvious that then, by the above assumption there is a constant such that ◻Lemma 8. Let and . Let . If there is a constant (not depending on ) such that
for all -linear forms and all positive integers , then for all -linear forms and all positive integers . Moreover, if and, for every , the cannot be improved (here and henceforth, this means that the cannot be replaced by any -sum).Proof. To simplify the notation, we can suppose .
Let us fix the last variables and work with -linear forms . Since
for all -linear forms , we know that there is a constant , such that for any fixed vectors , we have for all -linear forms . Then, there is a constant , such that for all -linear forms .Now let us show that the cannot be improved. In fact, in this case we have suprema, none of which can be improved. Otherwise there will exist , and such that
for all -linear forms and all . Using the Lemma 7, this would imply the existence of a constant such that for all -linear forms . Considering by the monotonicity of the norms we conclude that there is a constant such that for all -linear forms . But this is impossible due to the hypothesis . ◻In the next sections, using Lemma 7 and Lemma 8 , we obtain the super-critical versions of the Hardy-Littlewood inequalities presented in the introduction.
A first natural illustration of the usefulness of Lemma 7 and Lemma 8 leads us to an alternate proof of Proposition 6.3 in Pellegrino et al. 2017PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ DM & TEIXEIRA EV. 2017. A regularity principle in sequence spaces and applications. Bull Sci Math 141: 802-837.. In fact, if , it is well known that
for all bounded linear forms , if, and only if, Thus, for and such that , we invoke Lemma 7 and Lemma 8 to obtain:Proposition 9. (see Proposition 6.3 in Pellegrino et al. 2017PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ DM & TEIXEIRA EV. 2017. A regularity principle in sequence spaces and applications. Bull Sci Math 141: 802-837.) Let be such that We have
for all bilinear forms and all if, and only if, the exponents satisfyIn this section we are mainly interested in the case of -linear forms.
By Theorem 6 used for -linear forms we have
for all -linear forms and all positive integers , with and . Moreover, the supremum cannot be replaced by an -sum and is sharp; besides, the optimal exponent satisfying (8) fulfillsAs a consequence of Lemma 7 and Lemma 8, we complete the above result.
Proposition 10. Let and be such that and . The following assertions are equivalent:
-
There is a constant (not depending on ) such that
for every -linear form and all . -
The exponents satisfy
and
Proof. Since , by Theorem 3 there is a constant such that
for all bilinear forms if, and only if, and We combine this equivalence with the fact and , and then, we invoke Lemma 7 and Lemma 8 to conclude the proof. ◻Corollary 11. For all -linear forms and all , we have
if, and only if, and .The m-linear case
Now we use Lemma 7 and Lemma 8 to obtain super-critical versions of Hardy-Littlewood inequalities for -linear forms. Our main result is the following Theorem. Below, we use the notation to represent the smallest integer greater than to , i.e.,
Theorem 12. Let be an integer, , and . Then, there is a constant (not depending on ) such that
for every -linear form if, and only if,Moreover, the cannot be improved.Proof. The case is precisely (4) , so we shall assume . Since we have
and thus On the other hand we also have By (4) there is a constant such that for every -linear form if, and only if, By Lemma 8 with , and Lemma 7 we conclude the proof. ◻We finish this section with some super-critical results in the anisotropic setting, whose proofs we omit. We begin with a super-critical version of Theorem 5 :
Theorem 13. Let , , , and such that
and for all The following assertions are equivalent:-
There is a constant (not depending on ) such that
for all -linear forms and all -
The exponents satisfy
Analogously, using Lemma 7, Lemma 8 and Theorem 4 we have:
Theorem 14. Let and be such that
and for all , and The following assertions are equivalent:-
There is a constant (not depending on such that
for all -linear forms and all positive integers . -
and the inequality
is verified.
The next result shows that it is possible to avoid the condition , for all :
Theorem 15. Let and be such that
and with The following assertions are equivalent:-
There is a constant (not depending on such that
for all -linear forms and all positive integers . -
.
Proof. Suppose that holds and . In this case, Lemma 7 provides a constant such that
for all -linear forms and all positive integers . For any -linear form , we define an -linear form with the same rule of , but different domain . So, there is a constant such that for all -linear forms , and the exponents satisfyOn the other hand, replacing the unimodular -linear form of the Kahane-Salem-Zygmund inequality (see Lemma 6.1 in Albuquerque et al. 2014ALBUQUERQUE N, BAYART F, PELLEGRINO D & SEOANE-SEPÚLVEDA JB. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740.) in (10) , we obtain
Since this is valid for all , we conclude that and this is a contradiction. Hence . Finally, the fact that is a consequence of Lemma 8, because for all (recall that ).Finally, using Theorem 4 and Lemma 8 we prove that implies . ◻
Remark 16. It is worth mentioning that the above theorems are independent. For instance, if , , and , nothing can be inferred by Theorem 14 . However, using Theorem 15, we conclude that if and then there is a constant (not depending on such that
for all -linear forms and all positive integers if, and only if, .The following result was proved in Albuquerque & Rezende 2018ALBUQUERQUE N & REZENDE L. 2018. Anisotropic regularity principle in sequence spaces and applications. Comm Contemp Math 20: 1750087-1750100. (in Corollary 2):
Theorem 17. (see Corollary 2 in Albuquerque Rezende 2018) Let be a positive integer and and . Then, there is a constant (not depending on ) such that
for all -linear forms and all positive integers with for allAgain, Lemma 7 and Lemma 8 combined with the Kahane-Salem-Zygmund inequality (see Lemma 6.1 in Albuquerque et al. 2014ALBUQUERQUE N, BAYART F, PELLEGRINO D & SEOANE-SEPÚLVEDA JB. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740.) and Lemma 3.1 in Aron et al. 2017ARON R, NÚÑEZ-ALARCÓN D, PELLEGRINO D & SERRANO-RODRÍGUEZ D. 2017. Optimal exponents for Hardy-Littlewood inequalities for m-linear operators. Linear Algebra Appl 531: 399-422. give us the following super-critical version of the Theorem 17 :
Theorem 18. Let , , and , be such that
and for all . Then for all -linear forms and all positive integers , with and for all Moreover, , and the optimal exponents satisfying (12) are such that and the inequality is verified.
Remark 19. When and we recover Theorem 6 .
ACKNOWLEDGMENTS
The authors thank the reviewers for his/her careful reading and important suggestions that helped to improve the paper. D. Pellegrino is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant 307327/2017-5 and by Fundação de Apoio à Pesquisa do Estado da Paraíba (FAPESQ) Grant 2019/0014.
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2010 Mathematics Subject Classification: 47B37, 47B10, 11Y60.
- ALBUQUERQUE N, BAYART F, PELLEGRINO D & SEOANE-SEPÚLVEDA JB. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740.
- ALBUQUERQUE N & REZENDE L. 2018. Anisotropic regularity principle in sequence spaces and applications. Comm Contemp Math 20: 1750087-1750100.
- ARON R, NÚÑEZ-ALARCÓN D, PELLEGRINO D & SERRANO-RODRÍGUEZ D. 2017. Optimal exponents for Hardy-Littlewood inequalities for m-linear operators. Linear Algebra Appl 531: 399-422.
- CAVALCANTE W. 2018. Some applications of the regularity principle in sequence spaces. Positivity 22: 191-198.
- DIMANT V & SEVILLA-PERIS P. 2016. Summation of coefficients of polynomials on ℓp spaces. Publ Mat 60: 289-310.
- HARDY G & LITTLEWOOD JE. 1934. Bilinear forms bounded in space Quart J Math 5: 241-254.
- OSIKIEWICZ B & TONGE A. 2001. An interpolation approach to Hardy-Littlewood inequalities for norms of operators on sequence spaces. Linear Algebra Appl 331: 1-9.
- PAULINO D. 2019. Critical Hardy-Littlewood inequality for multilinear forms. Rend. Circ. Mat. Palermo, II. 69 (2020), 369-380.
- PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ DM & TEIXEIRA EV. 2017. A regularity principle in sequence spaces and applications. Bull Sci Math 141: 802-837.
- PRACIANO-PEREIRA T. 1981. On bounded multilinear forms on a class of ℓp spaces. J Math Anal Appl 81: 561-568.
- SANTOS J & VELANGA T. 2017. On the Bohnenblust-Hille inequality for multilinear forms. Results Math 72: 239-244.
Publication Dates
-
Publication in this collection
20 Feb 2023 -
Date of issue
2023
History
-
Received
21 Feb 2020 -
Accepted
20 Aug 2020