Abstract
The Bohnenblust-Hille polynomial and multilinear inequalities were proved in 1931 and the determination of exact values of their constants is still an open and challenging problem, pursued by various authors. The present paper briefly surveys recent attempts to attack/solve this problem; it also presents new results, like connections with classical results of the linear theory of absolutely summing operators, and new perspectives.
Keywords
Bohnenblust-Hille inequality; Hardy-Littlewood inequality; extreme points; Banach spaces
INTRODUCTION
Motivated by a question raised by P.J. Daniell concerning the existence of certain functions of bounded variation, Littlewood (193020 LITTLEWOOD JE. 1930. On bounded bilinear forms in an infinite number of variables. Quart J Math 1: 164-174.) proved that
for all bilinear forms and all positive integers (as usual, denotes with the norm and its canonical unit vectors). This inequality is now called Littlewood’s inequality. One year later, Bohnenblust and Hille (1931)9 BOHNENBLUST H AND HILLE E. 1931. On the absolute convergence of Dirichlet series. Ann Math 32: 600-622. extended Littlewood’s inequality to the multilinear framework by proving a key result to solve a long standing problem posed by H. Bohr (1913)10 BOHR H. 1913. Über die gleichmäßige Konvergenz Dirichletscher Reihen. Zeitschrift für reine und angewandte Mathematik, Bd 143., related to the convergence of Dirichlet series. Bohnenblust and Hille proved that the optimal constants satisfying
for all -linear forms are such that
Above and henceforth, as usual,
The case in (1) recovers Littlewood’s inequality. As a matter of fact Bohnenblust and Hille seemed to be more interested in the following variant of (1): there is a constant such that
for all -homogeneous polynomials of the form It is worth mentioning that the exponent in (1) and (2) is sharp.
Both multilinear and polynomial inequalities also hold for real scalars instead of complex scalars. From now on will denote or . The case of complex scalars is the original one, with strong connections with Analytic Number Theory and Dirichlet series; we mention Defant et al. (2011)13 DEFANT A, FRERICK L, ORTEGA-CERDÀ J, OUNAÏES M AND SEIP K. 2011. The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann Math 174: 485-497. and Bayart et al. (20146 BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. The Bohr radius of the n-dimensional polydisc is equivalent to square[(log n)/n]. Adv Math 264: 726-746.) for this line of research. The case of real scalars was just very recently explored but seems to be also very relevant as it can be seen in its applications in Quantum Information Theory, remarked by Montanaro (2012)22 MONTANARO A. 2012. Some applications of hypercontractive inequalities in quantum information theory. J Math Phys 53: 15.. It may sound surprising but, by now, the estimates of the constants are what really matters for the applications.
The original estimates obtained by Bohnenblust and Hille were
and since 2011 a series of papers have been dedicated to the investigation of the constants , . Up to now, according to Bayart et al. (2014)6 BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. The Bohr radius of the n-dimensional polydisc is equivalent to square[(log n)/n]. Adv Math 264: 726-746. and Nuñez-Alarcón (2013), the best known asymptotic estimates for these constants in the case of complex scalars are
where is the Euler-Mascheroni constant, can be arbitrarily chosen and depends on . It has been recently shown in Maia et al. (2017)21 MAIA M, NOGUEIRA T AND PELLEGRINO D. 2017. The Bohnenblust-Hille inequality for polynomials whose monomials have a uniformly bounded number of variables, to appear in Integral Equations and Operators Theory. DOI: 10.1007/s00020-017-2372-z. that under certain mild assumptions, is bounded. For real scalars, combining information from Bayart et al. (2014)6 BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. The Bohr radius of the n-dimensional polydisc is equivalent to square[(log n)/n]. Adv Math 264: 726-746., Campos et al. (2015)11 CAMPOS J, JIMÉNEZ-RODRÍGUEZ P, MUÑOZ-FERNÁNDEZ G, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2015. On the real polynomial Bohnenblust-Hille inequality. Lin Algebra Appl 465: 391-400. and Diniz et al. (2014)16 DINIZ D, MUÑOZ-FERNÁNDEZ G, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. Lower bounds for the constants in the Bohnenblust-Hille inequality: the case of real scalars. Proc Amer Math Soc 142: 575-580. we have
for some (small) .
Despite the huge recent advances in the theory (exponential estimates were improved to sublinear estimates and super-exponential estimates were improved to subpolynomial estimates), there are still many basic/simple open problems and new tools seem to be needed to a better understanding of the whole scenery. Below we list some basic open problems in this setting:
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Is the sequence bounded? This was conjectured to be true in Pellegrino and Teixeira (2017)25 PELLEGRINO D AND TEIXEIRA E. 2017. Towards sharp Bohnenblust-Hille constants, to appear in Comm Contemp Math. DOI:http://dx.doi.org/10.1142/S0219199717500298. but it seems to be far from being solved.
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Is the sequence increasing?
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Is for all positive integers
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What is the asymptotic growth of
Of course, a more ambitious problem is:
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What are the optimal values of and
One can also try to figure out a complete perspective by attacking the even more ambitious question in the multilinear setting:
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For each positive integers , what are the optimal values
Hardy and Littlewood (1934)17 HARDY GH AND LITTLEWOOD JE. 1934. Bilinear forms bounded in space [p,q]. Quart J Math 5: 241-254. extended Littlewood’s inequality to bilinear forms on spaces. The multilinear version of their result was obtained in Praciano-Pereira (1981)26 PRACIANO-PEREIRA T. 1981. On bounded multilinear forms on a class of lp spaces. J Mat Anal Appl 81: 561-568. for . The result is summarized as follows: there exists a constant such that, for all continuous -linear forms , and all positive integers ,
and the exponent is optimal. Further generalizations to the anisotropic settings were obtained in Albuquerque et al. (20142 ALBUQUERQUE N, BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740.) and the best known estimates for can be found in Araujo and Pellegrino (20144 ARAUJO G AND PELLEGRINO D. 2014. Lower bounds for the constants of the Hardy-Littlewood inequalities. Linear Algebra Appl 463: 10-15., 20175 ARAUJO G AND PELLEGRINO D. 2017. On the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities. Bull Braz Math Soc 48: 141-169.) and Cavalcante et al. (2016)12 CAVALCANTE W, NÚÑEZ-ALARCÓN D AND PELLEGRINO D. 2016. New lower bounds for the constants in the real polynomial Hardy-Littlewood inequality. Numer Funct Anal Optim 37: 927-937.. The case was recently explored in Dimant and Sevilla-Peris (2016)15 DIMANT V AND SEVILLA-PERIS P. 2016. Summation of coefficients of polynomials on lp spaces. Publ Mat 60: 289-310. and the constants involved were further explored in Albuquerque et al. (20171 ALBUQUERQUE N, ARAUJO G, MAIA M, NOGUEIRA T AND SANTOS J. 2017. Optimal Hardy-Littlewood inequalities uniformly bounded by a universal constant. arXiv:1609.03081.), Nunes (2017)23 NUNES AG. 2017. A new estimate for the constants of an inequality due to Hardy and Littlewood. Linear Algebra Appl 526: 27-34., among others.
The main goal of this paper is to survey some aspects of the Bohnenblust-Hille and Hardy-Littlewood inequalities and also present new results and perspectives. These inequalities have been exhaustively investigated in the recent years and several new results and new techniques have appeared. We are also interested in showing connections between these inequalities and some classical results of the theory of absolutely summing operators.
This paper is organized as follows. In the second section we show how the investigation of the geometry of the unit ball of the spaces of multilinear forms can be potentially useful to reach the optimal constants. In the third section we show how these kind of inequalities recover classical results of the linear theory of absolutely summing operators. In the final section we discuss the perspectives of the subject and new strategies to attack the problem.
THE MULTILINEAR BOHNENBLUST-HILLE CONSTANTS: GEOMETRIC APPROACHES
In this section we discuss how the geometry of the unit balls of Banach spaces is connected to the problem of finding optimal constants of the multilinear Bohnenblust-Hille inequality.
ESTIMATING NORMS
Given an -linear form is there a handy formula for If the answer was positive, then a definitive answer to the optimization problem of finding the optimal constants of the Bohnenblust-Hille inequality could be eventually achieved by a kind of Lagrange Multipliers approach. Unfortunately, to estimate the norm of a general multilinear form seems to be a quite complicated task. The following result (of independent interest) is a prediction:
Proposition 1.
Let be given by with . Then
(A)
if and and
(B)
otherwise.
Proof.
Note that
where is given by
We thus have
Hence, calculating is the same of maximizing the function
on Since is a convex function on a consequence of the Krein-Milman Theorem assures that the maximum of is attained on some extreme point of (this will be discussed in the depth in the next subsection) It is simple to verify that extreme points of (this result is used, for instance, by Diestel et al. 1995) are of the form with Denoting , , we have
Since , we can write , . Hence
By making we have
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Proof of (A):
We divide the proof of (A) in two cases:
First case. Suppose that and and
In this case, since and and , always exists and
if, and only if, or
Since , we have
and since
there is such that
Thus
Second case. Suppose that and
In this case there are real numbers such that does not exist. For these values of we can see that
or
For the values of such that exists, we proceed as in the first case; therefore we also obtain (6).
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Proof of (B).
We consider three cases:
Case 1. Suppose and with
From (4) we can observe that if and only if and thus
Case 2. Suppose , , ,
and
In this case, from (5) we also know that if and only if ; therefore
Case 3. We may have one of the following situations:
(1) and ;
(2) and ;
(3) and ;
(4) and
If we consider (1), can be written as one of the following expressions:
(a) ;
(b) ;
(c) ;
(d) .
We thus can write, in any case,
and, of course, we obtain the expression of (B).
If we consider (2) there is no loss of generality in supposing So, we get
and we consider two subcases.
Subcase 1. If or , then there is a such that does not exist. In this case, it is plain that
For other values of we have
and thus if and only if and . For these values of we have
or
We thus have again the expression given in (B).
Subcase 2. If and , then for all and exists for all thus we again obtain the expression of (B).
The situation (3) is similar to (2).
If we have (4) and we proceed as in the second case of (A). If and we are encompassed by Case 1 or Case 2 of (B). ∎
For real scalars, for the obvious reasons, the expression of the norm is less complicated; this result can be found in Jameson (1994)18 JAMESON GJO. 1994. A specific form of Grothendieck’s inequality for the two-dimensional case, with applications to C*-algebras. Proc Edinburgh Math Soc 37: 521-537.:
Proposition 2.
Let be given by with . Then
The above expressions of the norms of bilinear forms are somewhat prohibitive for further investigations following this vein. As we will see in the next section, the investigation of the geometry of the closed unit balls of is important for our goals and may overcome the difficulty of finding formulas for the norms of multilinear forms.
EXTREME POINTS AND OPTIMIZATION
Given a vector space and a convex set , a vector is called extreme point of when with implies . Extreme points are important for optimization of continuous and convex functions for a simple reason: their maximum are attained in extreme points as we shall see below.
Let us first state the Minkowski/Krein-Milman Theorem. It asserts that given a convex and compact subset , then . Here denotes the set of all extreme points of and denotes the convex hull of .
If is a convex continuous function, i.e., a continuous function such that
then its maximum is attained in an extreme point . In fact, suppose that is a point where the maximum is attained; the Minkowski/Krein-Milman Theorem asserts that there are such that
with and If the maximum of is not attained in any extreme point, then
a contradiction. The same happens with multilinear forms in finite dimensional normed spaces as the next result (which seems to be folklore) asserts:
Proposition 3.
Let be a positive integer, be a finite dimensional normed space and be an -linear form. Then, denoting by its closed unit ball, we have
Proof.
It suffices to prove that there are such that
Let us suppose that (7) is not true. Since is compact, there exist such that . By the Minkowski/Krein-Milman Theorem, we have
and thus
with , and for all . Hence
a contradiction. ∎
The following lines illustrate how the previous results can be useful to our goals.
We want to solve the optimization problem
for (recall that denotes the space of all -linear forms and is its closed unit ball). Since is convex and compact and the function
given by
is convex, the Minkowski/Krein-Milman Theorem tells us that the supremum of is attained in some extreme point of Hence, we are guided to investigate the geometry of
GEOMETRY OF UNIT BALL OF : A MYSTERY TO BE SOLVED
The geometry of the unit ball of is, in general, unknown. This section is entirely devoted to characterizing the geometry of the unit ball of We start off with four elementary lemmata; the proofs are omitted.
Lemma 4.
Let and be such that and
If , then or .
Lemma 5.
Let and . Then
if, and only if,
Lemma 6.
Let and . Then
if, and only if,
Theorem 7.
The extreme points of the closed unit ball of are
Proof.
For the sake of simplicity we shall denote by along this proof. Let be given by . By symmetry, it suffices to consider the following cases, with :
(1) ;
(2) ;
(3) ;
(4) .
Since , we know that and are not bigger than .
Case (1). If let . Defining
we have and . Thus, is not an extreme point. If , we can suppose . Thus, if there are such that , say,
we have Since , we conclude that . Note that if , then or is bigger than . Estimating and we conclude that and the same happens to ; therefore . The same argument shows us that . Thus, is an extreme point.
Case (2). Note that
Let and defining
we conclude that and . Thus, is not an extreme point.
Case (3). By Proposition 2, we have
Note that
Let us consider two subcases:
(3A) ;
(3B) .
If (3A) happens, then . Defining and
we have and . Thus is not extreme point.
If (3B) happens, then . Defining and
we have and . Thus is not extreme point.
Case (4). We consider four subcases:
(4A) ;
(4B) ;
(4C) ;
(4D) .
If (4A) happens, then . Considering and defining
we have and . Thus is not an extreme point.
If (4B) happens, then . Considering and defining
we have and . Thus, again, is not an extreme point.
If (4C) happens we can assume and . Note that by Proposition 2,
or
We shall consider just (8) because (9) is similar. If we have (8) then, by Lemma 5, there are two possibilities:
(4CA) and
(4CB) and
We shall first prove that if
then is not an extreme point. Let us first suppose (4CA).
If we can assume because the other cases are analogous. We thus have two possibilities:
(4CAA)
(4CAB)
Let us first consider (4CAA):
Since , we have and . We consider two cases:
(4CAAA) and ;
(4CAAB) and .
If (4CAAA) happens, since and we conclude that
i.e.,
and thus, by Proposition 2, . Considering and defining
we have and . Thus is not an extreme point.
If (4CAAB) happens, since and using that and we have
and, by then by Proposition 2,
We have two possibilities:
(4CAABA)
(4CAABB)
If (4CAABA) happens, we choose and define
and by Lemma 4, we conclude that and . Hence, once more, is not an extreme point.
If (4CAABB) happens, we can write
Since , it follows that
If , then
Note that
Since , it follows that . Considering and defining
we have and . Hence, is not an extreme point.
If , then . Considering and defining
we conclude that and . Thus is not a extreme point.
Now let us prove (4CAB). Since , then
If and , then
Hence
Considering and defining
we conclude that and . Thus is not an extreme point.
If and , then we shall proceed as in the case (4CAAB) to observe that
is not an extreme point.
So, it remains to look for extreme points in the case (4C) when
In this case we can write
Since , we have If , is not an extreme point. Let us show that when the bilinear form given by (10) is an extreme point.
Suppose that there exist such that . Denoting
we have
Since , it follows that . A similar argument tells us that . We claim that if , then or . Note that
and
In a similar fashion,
and so on.
So, let us first suppose . We may have or .
If , then and . Therefore,
and thus , a contradiction.
If , then we may have:
(P1) and ;
(P2) and ;
(P3) and ;
(P4) and .
If (P1) holds, then , and . Thus , and . When , by Lemma 5, we have
When , by Lemma 6, we have
If (P2) holds, then , and . Thus and . By Lemma 6, we have
If (P3) holds, then
and thus .
If we have (P4), then and . When , by Lemma 5, we have
When , by Lemma 6, we have
Now, let us suppose . We may have or . If , then
and hence , a contradiction. If , we may have:
(K1) and ;
(K2) and ;
(K3) and ;
(K4) and .
If (K1) happens, then , and . When , by Lemma 5, we have
When , by Lemma 6, we have
If (K2) occurs, then e . Therefore,
and .
If we have (K3), then
and so
If (K4) happens, then and . When , by Lemma 5, we have
When , by Lemma 6, we have
The case (4CB) is analogous to (4CA) and (4D) is similar to (4C). ∎
Using the above characterization of extreme points and the Minkowski/Krein-Milman Theorem we have an alternative proof that the optimal constants of the Littlewood’s inequality is This geometric approach will be later commented in the final section.
Remark 8.
Theorem 7 was independently proved in Kim (In Press19 KIM SG. IN PRESS. The geometry of ℒ(2ℓ∞2). Kyungpook Math J.).
CONNECTIONS WITH LINEAR ABSOLUTELY SUMMING OPERATORS
The aim of this section is to highlight the connections between the Bohnenblust-Hille/Hardy-Littlewood inequalities and the theory of absolutely summing operators. We show how recent results of the theory of Bohnenblust-Hille and Hardy-Littlewood inequalities are connected to classical results of the linear theory of absolutely summing operators. The results are essentially applications of recent contributions from Pellegrino et al. (In Press24 PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ D AND TEIXEIRA E. IN PRESS. Regularity principle in sequence spaces and applications. Bull Sci Math.).
Let , be Banach spaces and be real numbers; let denote the topological dual of and be the closed unit ball of For all , the real number denotes its conjugate number, i.e.,
We recall that a continuous linear operator is absolutely -summing if whenever where
which is a Banach space when endowed with the norm
The next result seems to be folklore.
Proposition 9.
The following assertions are equivalent for :
(i) The canonical inclusion map is absolutely -summing with constant
(ii)
for all continuous bilinear forms .
Proof.
(ii)(i). Let According to Diestel et al. (1995)14 DIESTEL J, JARCHOW H AND TONGE A. 1995. Absolutely summing operators. Cambridge Studies in Advanced Mathematics 43. there is an isometric isomorphism given by so there is a continuous linear operator such that for all Then
Recalling that isometrically, there is a bilinear form such that
(i)(ii). It is just to use the same isometric isomorphisms in the inverse direction. ∎
In Pellegrino et al. (In Press) it was proved that if and are such that
then
for all continuous bilinear forms if, and only if,
It is also well known that if
then then the inequality (11) is always impossible. So, by Proposition 9 we conclude that if , then (with of course) is absolutely -summing if and only if
i.e.,
We thus recover the following inequalities of Bennett (1977)8 BENNETT G. 1977. Schur multipliers. Duke Math J 44: 603-639. regarding inclusion summing norms:
Theorem 10.
Consider the canonical inclusion (with of course). If it is absolutely -summing if, and only if,
When we recover results of Bennett (19737 BENNETT G. 1973. Inclusion mappings between lp spaces. J Functional Anal 13: 20-27.):
Theorem 11.
Consider the inclusion (with of course). If it is absolutely -summing if and only if
Remark 12.
Our results also imply that when the optimal value of the constant for the case of real scalars is
CONCLUSIONS
Despite the progresses obtained in the last years there seems to be a long way until the Bohnenblust-Hille and Hardy-Littlewood inequalities are fully understood. In view of the analytic difficulties, a computational approach is reasonable alternative. In fact, some attempts in this direction can be seen in Araujo et al. (20173 ARAUJO G, JIMÉNEZ-RODRÍGUEZ P, MUÑOZ-FERNÁNDEZ G AND SEOANE-SEPÚLVEDA J. 2017. Equivalent norms in polynomial spaces and applications. J Math Anal Appl 445: 1200-1220.) and Cavalcante et al. (2016)12 CAVALCANTE W, NÚÑEZ-ALARCÓN D AND PELLEGRINO D. 2016. New lower bounds for the constants in the real polynomial Hardy-Littlewood inequality. Numer Funct Anal Optim 37: 927-937., but these approaches are restricted to very particular cases. A recent result of Pellegrino and Teixeira (2017)25 PELLEGRINO D AND TEIXEIRA E. 2017. Towards sharp Bohnenblust-Hille constants, to appear in Comm Contemp Math. DOI:http://dx.doi.org/10.1142/S0219199717500298. shows that a particular instance of the Bohnenblust-Hille inequality can be re-written as an algorithm, opening some possibilities for further computational investigations.
Since 2012, the main advances in the estimates of the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities were mainly obtained by interpolation, i.e., a kind of Hölder-type inequality for mixed spaces. However, as remarked in Pellegrino and Teixeira (2017)25 PELLEGRINO D AND TEIXEIRA E. 2017. Towards sharp Bohnenblust-Hille constants, to appear in Comm Contemp Math. DOI:http://dx.doi.org/10.1142/S0219199717500298. this seems to be still a sub-optimal approach. Several evidences were collected to support this claim and the notions of entropy and complexity introduced in Pellegrino and Teixeira (2017)25 PELLEGRINO D AND TEIXEIRA E. 2017. Towards sharp Bohnenblust-Hille constants, to appear in Comm Contemp Math. DOI:http://dx.doi.org/10.1142/S0219199717500298. may be helpful in this direction.
It is our belief that the geometry of the unit ball of although very hard and delicate, should be investigated in depth and an eventual advance in this direction, combined with the Minkowski/Krein-Milman Theorem could be an effective approach. Similarly, the investigation of the extreme points of the closed unit ball of seems to be the best option to provide a final answer to the questions related to the optimal constants of the Hardy-Littlewood inequalities.
ACKNOWLEDGMENTS
The authors thank the two anonymous referees for important suggestions. W.V. Cavalcante is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and D.M. Pellegrino is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) - Brasil, Grant 302834/2013-3.
References
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1ALBUQUERQUE N, ARAUJO G, MAIA M, NOGUEIRA T AND SANTOS J. 2017. Optimal Hardy-Littlewood inequalities uniformly bounded by a universal constant. arXiv:1609.03081.
-
2ALBUQUERQUE N, BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J Funct Anal 266: 3726-3740.
-
3ARAUJO G, JIMÉNEZ-RODRÍGUEZ P, MUÑOZ-FERNÁNDEZ G AND SEOANE-SEPÚLVEDA J. 2017. Equivalent norms in polynomial spaces and applications. J Math Anal Appl 445: 1200-1220.
-
4ARAUJO G AND PELLEGRINO D. 2014. Lower bounds for the constants of the Hardy-Littlewood inequalities. Linear Algebra Appl 463: 10-15.
-
5ARAUJO G AND PELLEGRINO D. 2017. On the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities. Bull Braz Math Soc 48: 141-169.
-
6BAYART F, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. The Bohr radius of the n-dimensional polydisc is equivalent to square[(log n)/n]. Adv Math 264: 726-746.
-
7BENNETT G. 1973. Inclusion mappings between lp spaces. J Functional Anal 13: 20-27.
-
8BENNETT G. 1977. Schur multipliers. Duke Math J 44: 603-639.
-
9BOHNENBLUST H AND HILLE E. 1931. On the absolute convergence of Dirichlet series. Ann Math 32: 600-622.
-
10BOHR H. 1913. Über die gleichmäßige Konvergenz Dirichletscher Reihen. Zeitschrift für reine und angewandte Mathematik, Bd 143.
-
11CAMPOS J, JIMÉNEZ-RODRÍGUEZ P, MUÑOZ-FERNÁNDEZ G, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2015. On the real polynomial Bohnenblust-Hille inequality. Lin Algebra Appl 465: 391-400.
-
12CAVALCANTE W, NÚÑEZ-ALARCÓN D AND PELLEGRINO D. 2016. New lower bounds for the constants in the real polynomial Hardy-Littlewood inequality. Numer Funct Anal Optim 37: 927-937.
-
13DEFANT A, FRERICK L, ORTEGA-CERDÀ J, OUNAÏES M AND SEIP K. 2011. The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann Math 174: 485-497.
-
14DIESTEL J, JARCHOW H AND TONGE A. 1995. Absolutely summing operators. Cambridge Studies in Advanced Mathematics 43.
-
15DIMANT V AND SEVILLA-PERIS P. 2016. Summation of coefficients of polynomials on lp spaces. Publ Mat 60: 289-310.
-
16DINIZ D, MUÑOZ-FERNÁNDEZ G, PELLEGRINO D AND SEOANE-SEPÚLVEDA J. 2014. Lower bounds for the constants in the Bohnenblust-Hille inequality: the case of real scalars. Proc Amer Math Soc 142: 575-580.
-
17HARDY GH AND LITTLEWOOD JE. 1934. Bilinear forms bounded in space [p,q]. Quart J Math 5: 241-254.
-
18JAMESON GJO. 1994. A specific form of Grothendieck’s inequality for the two-dimensional case, with applications to C*-algebras. Proc Edinburgh Math Soc 37: 521-537.
-
19KIM SG. IN PRESS. The geometry of Kyungpook Math J.
-
20LITTLEWOOD JE. 1930. On bounded bilinear forms in an infinite number of variables. Quart J Math 1: 164-174.
-
21MAIA M, NOGUEIRA T AND PELLEGRINO D. 2017. The Bohnenblust-Hille inequality for polynomials whose monomials have a uniformly bounded number of variables, to appear in Integral Equations and Operators Theory. DOI: 10.1007/s00020-017-2372-z.
-
22MONTANARO A. 2012. Some applications of hypercontractive inequalities in quantum information theory. J Math Phys 53: 15.
-
23NUNES AG. 2017. A new estimate for the constants of an inequality due to Hardy and Littlewood. Linear Algebra Appl 526: 27-34.
-
24PELLEGRINO D, SANTOS J, SERRANO-RODRÍGUEZ D AND TEIXEIRA E. IN PRESS. Regularity principle in sequence spaces and applications. Bull Sci Math.
-
25PELLEGRINO D AND TEIXEIRA E. 2017. Towards sharp Bohnenblust-Hille constants, to appear in Comm Contemp Math. DOI:http://dx.doi.org/10.1142/S0219199717500298.
-
26PRACIANO-PEREIRA T. 1981. On bounded multilinear forms on a class of lp spaces. J Mat Anal Appl 81: 561-568.
Publication Dates
-
Publication in this collection
01 Feb 2018 -
Date of issue
2019
History
-
Received
25 May 2017 -
Accepted
29 June 2017