Hölder's inequality: some recent and unexpected applications
N. Albuquerque, G. Araujo, D. Pellegrino, J. Seoane-Sepulveda
aa r X i v : . [ m a t h . F A ] M a r H ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTEDAPPLICATIONS
N. ALBUQUERQUE, G. ARA ´UJO, D. PELLEGRINO, AND J.B. SEOANE-SEP ´ULVEDA
Abstract.
H¨older’s inequality, since its appearance in 1888, has played afundamental role in Mathematical Analysis and it is, without any doubt, oneof the milestones in Mathematics. It may seem strange that, nowadays, it keepsresurfacing and bringing new insights to the mathematical community. In thisexpository article we show how a variant of H¨older’s inequality (although well-known in PDEs) was essentially overlooked in Functional Analysis and has hada crucial (and in some sense unexpected) influence in very recent and majorbreakthroughs in Mathematics. Some of these recent advances appeared in2012-2014 and include the theory of Dirichlet series, the famous Bohr radiusproblem, certain classical inequalities (such as Bohnenblust–Hille or Hardy–Littlewood), or even Mathematical Physics.
Contents
1. Introduction 12. Motivation: some interpolative puzzles 23. H¨older’s inequality revisited 54. Some useful inequalities 84.1. Khinchine’s inequality 104.2. Kahane–Salem–Zygmund’s inequality 114.3. A corollary to Minkowski’s inequality 115. Recent “unexpected” applications to classical problems 125.1. The Bohnenblust–Hille inequality with subpolynomial constants 125.2. Quantum Information Theory 165.3. Power series and the Bohr radius problem 175.4. Hardy–Littlewood’s inequality constants 205.5. Separately summing operators 21References 211.
Introduction
When Leonard James Rogers (1862-1933) and Otto H¨older (1859-1937) discov-ered, independently, the famous inequality that (nowadays) holds H¨older’s name
Mathematics Subject Classification.
Key words and phrases.
H¨older’s inequality, Minkowki’s inequality, interpolation, Bohr ra-dius, Quantum Information Theory, Hardy-Littlewood’s inequality, Bohnenblust-Hille’s inequality,Khinchine’s inequality, Kahane-Salem-Zygmund’s inequality, absolutely summing operators.D. Pellegrino and J.B. Seoane-Sep´ulveda are supported by CNPq Grant 401735/2013-3 - PVE- Linha 2. (1889, [39]), they could have never imagined that, at that precise moment, they hadjust started a “revolution” in Functional Analysis (we refer to [42] for a detailedand historical exposition). This tool is a fundamental inequality between integralsand an indispensable tool for the study of, among others, L p spaces. Let us recallthe classical L p version of this inequality. Theorem 1.1 (H¨older’s inequality, 1889) . Let (Ω , Σ , µ ) be a measure space and let p, q ∈ [1 , ∞ ] with /p + 1 /q = 1 (H¨older’s conjugates). Then, for all measurablereal or complex valued functions f and g on Ω , Z | f g | dµ ≤ (cid:18)Z | f | p dµ (cid:19) /p (cid:18)Z | g | q dµ (cid:19) /q . If one has p, q ∈ (1 , ∞ ), f ∈ L p ( µ ), and g ∈ L q ( µ ), then this inequality be-comes an equality if and only if | f | p and | g | q are linearly dependent in L ( µ ).When one has p = q = 2 we recover a form of the Cauchy-Schwarz inequality (orthe Cauchy-Bunyakovsky-Schwarz inequality). Also, H¨older’s inequality is used toprove Minkowski’s inequality (the triangle inequality for L p spaces) and to establishthat L q ( µ ) is the dual space of L p ( µ ) for p ∈ [1 , ∞ ). Of course, we are all familiarwith these classical applications of H¨older’s inequality.As it happens to almost every important result in mathematics, several exten-sions and generalizations of it have appeared along the time; and in the case ofH¨older’s inequality, this is not different. One of the extensions is the variant ofH¨older’s inequality for mixed L p spaces. This inequality appeared in 1961, in thework of A. Benedek and R. Panzone [8]. Mixed L p spaces may be seen as a pureexercise of abstraction of the original notion of L p spaces, but as a matter of factwe shall show that the theory developed in [8] plays a crucial role in applications toquite different frameworks; it is intriguing that, although widely known (the paper[8] has more than 100 citations, mainly related to PDEs; we refer, for instance to[20, 34]) it was overlooked in important fields of mathematics. This gap began to befilled in 2012-2013, when H¨older’s inequality for mixed L p spaces was re-discoveredas an interpolation-type result and we shall show that different fields of Mathe-matics and even of Physics were positively influenced. This expository paper isarranged as follows. Section 2 presents some motivation to illustrate the subjectof this article. Section 3 is devoted to the aforementioned variant of H¨older’s in-equality (H¨older’s inequality for mixed sums) with a short proof. This result wasonly written in a proper and organized fashion in 1961 ([8]) but, as it will be clearalong this paper, at least in the topics gathered here (Functional Analysis, ComplexAnalysis and Quantum Information Theory) it was surely not been taken advan-tage of before 2012. Our approach is quite different from the one employed in [8]and we shall follow the lines of [7]. Section 4 will recall several useful inequalitiesthat we shall need and Section 5 is devoted to the recent applications of H¨older’sinequality for mixed sums in Functional Analysis and Quantum Information The-ory, culminating with the solution of a classical problem from Complex Analysis:the Bohr radius problem. Applications to the improvement of the constants of theHardy–Littlewood inequality and separately summing operators are also given.2. Motivation: some interpolative puzzles
As a motivation to the subject treated here, let us suppose that we have thefollowing two inequalities at hand, for certain complex scalar matrix ( a ij ) Ni,j =1 : ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 3 (2.1) N X i =1 N X j =1 | a ij | ≤ C and N X j =1 N X i =1 | a ij | ! ≤ Cfor some constant C > N .How can one find an optimal exponent r and a constant C > N X i,j =1 | a ij | r r ≤ C for all positive integers N ? Moreover, how can one get a good (small) constant C ?This question (at least concerning the exponent r can be solved in no less thantwo ways: interpolation and H¨older’s inequality).First note that, by using a consequence of Minkowski’s inequality (see [35]), weknow that(2.2) N X i =1 N X j =1 | a ij | ≤ N X j =1 N X i =1 | a ij | ! ≤ C . If we use H¨older’s inequality twice, we proceed as follows: N X i,j =1 | a ij | = N X i =1 N X j =1 | a ij | | a ij | ≤ N X i =1 N X j =1 | a ij | N X j =1 | a ij | ≤ N X i =1 N X j =1 | a ij | N X i =1 N X j =1 | a ij | = N X i =1 N X j =1 | a ij | N X i =1 N X j =1 | a ij | ≤ C . By complex interpolation (see [9]) the solution is shorter; essentially we havetwo mixed inequalities with exponents (1 ,
2) in equation (2.1) and (2 ,
1) in equation(2.2). By interpolating these exponents with θ = θ = 1 / / , /
3) with constant C.The optimality of the exponent 4 / ALBUQUERQUE, ARA´UJO, PELLEGRINO, AND SEOANE N X σ ( i )=1 N X σ ( j )=1 N X σ ( k )=1 | a ijk | ≤ Cfor all bijections σ : { i, j, k } → { i, j, k } and all N. How can we find an optimalexponent r and a constant C such that N X i,j,k =1 | a ijk | r r ≤ C for every N ?The search for good constants dominating the respective inequalities is highlyimportant for applications (see Section 5) and has an extra ingredient when we areusing the interpolative approach: the main point is that different interpolationsmay result in the same exponent, but the constants involved differ. Thus, we mustinvestigate what exponents we shall use to interpolate. More precisely, as we willsee in Section 5, the Bohnenblust–Hille inequality for 3-linear forms asserts thatthere is a constant C ≥ T : ℓ N ∞ × ℓ N ∞ × ℓ N ∞ → K , N X i ,i ,i =1 (cid:12)(cid:12) T ( e i , e i , e i ) (cid:12)(cid:12) ≤ C k T k , and all N , where k T k := sup k z (1) k =1 ,..., k z ( m ) k =1 (cid:12)(cid:12)(cid:12) T (cid:16) z (1) , . . . , z ( m ) (cid:17)(cid:12)(cid:12)(cid:12) for all m ∈ N . However, the exponent 3 / N X i =1 N X i =1 N X i =1 (cid:12)(cid:12) T ( e i , e i , e i ) (cid:12)(cid:12) q ! q q q q q ≤ C k T k , with ( q , q , q ) = (1 , , , (2 , ,
2) and (2 , , q , q , q ) = (cid:18) , , (cid:19) , (cid:18) , , (cid:19) and (cid:18) , , (cid:19) and the last procedure provides quite better constants. This is a simple illustrationof the core of the new advances that lead to the results reported in this expositorypaper.The theory of L p spaces with mixed norms seems to have been created in 1961,[8], including a H¨older inequality in this framework. However, as we describe alongthis paper the full strength of this inequality was overlooked and very recentlyimportant advances in different fields of Mathematics and Physics were achievedwith the help of this H¨older inequality (also called interpolative approach). ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 5 H¨older’s inequality revisited
Essentially, the simplest version of the H¨older inequality asserts that if 1 /p +1 /q = 1 and ( a j ) ∈ ℓ p , ( b j ) ∈ ℓ q then ( a j b j ) ∈ ℓ . In this section we present avariation of this result, which was apparently overlooked in Functional Analysis(but not in PDEs) in the last decades. This variant is a key result of a numberof important recent advances in Mathematical Analysis and Mathematical Physicsthat appeared in the last few years.The previous result may have been seen as a variant of the following generalH¨older’s inequality presented in the classical paper [8] on mixed norms in L p spaces.We shall now work with L p ( N ) = ℓ p , since it is the case we are interested in. Weneed to recall some useful multi-index notation: for a positive integer m and anon-void subset D ⊂ N we denote the set of multi-indices i = ( i , . . . , i m ), witheach i k ∈ D , by M ( m, D ) := { i = ( i , . . . , i m ) ∈ N m ; i k ∈ D, k = 1 , . . . , m } = D m . We also denote M ( m, n ) := M ( m, { , , . . . , n } ) . For p = ( p , . . . , p m ) ∈ [1 , ∞ ) m , and a Banach space X , let us consider the space ℓ p ( X ) := ℓ p ( ℓ p ( . . . ( ℓ p m ( X )) . . . )) , namely, a vector matrix ( x i ) i ∈M ( m, N ) ∈ ℓ p ( X ) if, and only if, ∞ X i =1 ∞ X i =1 . . . ∞ X i m − =1 ∞ X i m =1 k x i k p m X ! pm − pm pm − pm − . . . p p p p p < ∞ . When X = K , we just write ℓ p instead of ℓ p ( K ).Also, we deal with the coordinatewise product of two scalar matrices a =( a i ) i ∈M ( m,n ) and b = ( b i ) i ∈M ( m,n ) , i.e. , ab := ( a i b i ) i ∈M ( m,n ) . The following result seems to be first observed by A. Benedek and R. Panzone(see [8]):
Theorem 3.1 (H¨older’s inequality for mixed ℓ p spaces) . Let r , q (1) , . . . , q ( N ) ∈ [1 , ∞ ] m such that r j = 1 q j (1) + · · · + 1 q j ( N ) , j ∈ { , , . . . , m } and let a k , k = 1 , . . . , N be scalar m -square matrix. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N Y k =1 a k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r ≤ N Y k =1 k a k k q ( k ) . Remember that, the previous inequality means the following:
ALBUQUERQUE, ARA´UJO, PELLEGRINO, AND SEOANE n X i =1 . . . n X i m =1 | a i · a i · . . . · a N i | q m ! qm − qm . . . q q q ≤ N Y k =1 n X i =1 . . . n X i m =1 | a i | q m ( k ) ! qm − k ) qm ( k ) . . . q k ) q k ) q k ) , Using the above result we are able to recover the interpolative inequality from[2–4,7] (Theorem 3.2 below), that we can also, in some sense, call H¨older’s inequalityfor multiple exponents. Under the point of view of interpolation theory it is nota complicated result but, just in 2013, it began to be used in all its full strength.Its applications (in different fields) are impressive, as we shall illustrate in theremaining of the paper. Just before that, for a positive real number θ , let us define a θ := (cid:0) a θ i (cid:1) i ∈M ( m,n ) . It is straightforward to see that (cid:13)(cid:13) a θ (cid:13)(cid:13) q /θ = k a k θ q , where q /θ := ( q /θ, . . . , q m /θ ). Theorem 3.2 (H¨older’s inequality for multiple exponents -interpolative approach-) . Let m, n, N be positive integers and q , q (1) , . . . , q ( N ) ∈ [1 , ∞ ) m be such that (cid:16) q , . . . , q m (cid:17) belongs to the convex hull of (cid:16) q ( k ) , . . . , q m ( k ) (cid:17) , k = 1 , . . . , N . Then for all scalarmatrix a = ( a i ) i ∈M ( m,n ) , k a k q ≤ N Y k =1 k a k θ k q ( k ) , i.e. , n X i =1 . . . n X i m =1 | a i | q m ! qm − qm . . . q q q ≤ N Y k =1 n X i =1 . . . n X i m =1 | a i | q m ( k ) ! qm − k ) qm ( k ) . . . q k ) q k ) q k ) θ k , where θ k are the coordinates of (cid:16) q ( k ) , . . . , q m ( k ) (cid:17) on the convex hull.Proof. For j = 1 , . . . , m we have1 q j = θ q j (1) + . . . + θ N q j ( N ) = 1 q j (1) /θ + . . . + 1 q j ( N ) /θ N . Since (cid:13)(cid:13) a θ k (cid:13)(cid:13) q ( k ) /θ k = k a k θ k q ( k ) , by the H¨older inequality for mixed ℓ p spaces weconlude that k a k q = (cid:13)(cid:13) a θ + ··· + θ N (cid:13)(cid:13) q = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N Y k =1 a θ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) q ≤ N Y k =1 (cid:13)(cid:13) a θ k (cid:13)(cid:13) q ( k ) /θ k = N Y k =1 k a k θ k q ( k ) . (cid:3) ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 7 For the sake of completeness of this article, we would also like to present thefollowing proof, which is based on interpolation. (Interpolative Approach).
We just follow the lines of [2, Proposition 2.1]. Proceed-ing by induction on N and using that, for any Banach space X and θ ∈ [0 , ℓ r ( X ) = [ ℓ p ( X ) , ℓ q ( X )] θ , with r i = θp i + − θq i , for i = 1 , . . . , m (see [9]). If1 q i = θ q i (1) + · · · + θ N q i ( N ) , with P Nk =1 θ k = 1 and each θ k ∈ [0 , q i = θ q i (1) + 1 − θ p i , setting 1 p i = α q i (2) + · · · + α N q i ( N ) , and α j = θ j − θ , for i = 1 , . . . , m and j = 2 , . . . , N . So α j ∈ [0 ,
1] and P Nj =2 α j = 1. Therefore, bythe induction hypothesis, we conclude that k a k q ≤ k a k θ q (1) · k a k − θ p ≤ k a k θ q (1) · N Y j =2 k a k α j q ( j ) − θ = N Y k =1 k a k θ k q ( k ) . (cid:3) Combining the previous result with Minkowski’s inequality we have a very usefulinequality (see [7, Remark 2.2]):
Corollary 3.3.
Let m, n be positive integers, ≤ k ≤ m and ≤ s ≤ q . Then forall scalar matrix ( a i ) i ∈M ( m,n ) , X i ∈M ( m,n ) | a i | ρ ρ ≤ Y S ∈P k ( m ) X i S X i b S | a i | q sq s · ( mk ) , where ρ := msqkq + ( m − k ) s and P k ( m ) stands for the set of subsets S ⊆ { , . . . , m } with card( S ) = k . The above corollary shows that Blei’s inequality (see Corollary 3.4 below) is justa very particular case of a huge family of similar inequalities. For our purposes,the crucial point is that the use of Blei’s inequality is far from being a good optionto obtain good estimates for the constants of the Bohnenblust–Hille and relatedinequalities. Just to illustrate the strength of Theorem 3.2 and Corollary 3.3, wepresent here a very simple proof (see [7]) of Blei’s inequality.
ALBUQUERQUE, ARA´UJO, PELLEGRINO, AND SEOANE
Corollary 3.4 (Blei’s inequality - Defant, Popa, Schwarting approach, [25]) . Let A and B be two finite non-void index sets. Let ( a ij ) ( i,j ) ∈ A × B be a scalar matrixwith positive entries, and denote its columns by α j = ( a ij ) i ∈ A and its rows by β i = ( a ij ) j ∈ B . Then, for q, s , s ≥ with q > max( s , s ) we have X ( i,j ) ∈ A × B a w ( s ,s ) ij w ( s ,s ≤ X i ∈ A k β i k s q ! f ( s ,s s X j ∈ B k α j k s q f ( s ,s s , with w : [1 , q ) → [0 , ∞ ) , w ( x, y ) := q ( x + y ) − qxyq − xy ,f : [1 , q ) → [0 , ∞ ) , f ( x, y ) := q x − qxyq ( x + y ) − qxy . Proof.
Let us consider the exponents( q, s ) , ( s , q )and ( θ , θ ) = ( f ( s , s ) , f ( s , s )) . Note that ( w ( s , s ) , w ( s , s )) is obtained by interpolating ( q, s ) and ( s , q ) with θ , θ , respectively. Then, from Theorem 3.2, we have X ( i,j ) ∈ A × B a w ( s ,s ) ij w ( s ,s ≤ X i ∈ A k β i k s q ! f ( s ,s s X i ∈ A k β i k qs ! f ( s ,s q . Now, since q > s we just need to use Propositon 4.6 to change the order of thelast sum. (cid:3) We invite the interest reader to compare the above proof with the proof presentedin [25, pages 226-227], in which the classical H¨older’s inequality is needed severaltimes. 4.
Some useful inequalities
The main recent advances presented here are direct or indirect consequence ofthe improvements obtained in the polynomial and multilinear Bohnenblust–Hilleinequalities (and these improvements were obtained by using the theory of mixed L p spaces and the core of the results lie in the variant of H¨older’s inequality (The-orem 3.2). However we also need three other important ingredients: the Khinchineinequality (and its version for multiple sums), Kahane–Salem–Zygmund’s inequalityin its polynomial and multilinear versions and a variant of Minkowski’s inequality.Before that, let us provide a brief account on polynomials and multilinear operators,that shall be needed in the remaining of the article.Polynomials in Banach spaces (at least for complex scalars) are of fundamentalimportance in the theory of Infinite Dimensional Holomorphy (see [30, 45]). Ingeneral the theory of polynomials and multilinear operators between normed spaceshas its importance in different areas of Mathematics, from Number Theory, orDirichlet series, to Functional Analysis.In this section we recall the concepts of polynomials and multilinear operatorsbetween Banach spaces and some “folkloric results”, that will be needed here. ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 9 If E , . . . , E m , and F are vector spaces, a m -linear operator T : E ×· · ·× E m → F is a map that is linear in each coordinate separately. When E = · · · = E m = E we say that T is symmetric if T ( x σ (1) , . . . , x σ ( m ) ) = T ( x , . . . , x m ) for all bijections σ : { , . . . , m } → { , . . . , m } . If E, F are vector spaces, a m -homogeneous polynomial is a map P : E → F such that P ( x ) = T ( x, . . . , x )for some m -linear operator T : E × · · ·× E → F. Continuity is defined is the obviousfashion.Fixed
E, F, E , . . . , E m , the spaces of continuous m -homogeneous polynomialsfrom E to F are represented by P ( m E ; F ) and the space of continuous multilinearoperators from E × · · · × E m to F is denoted by L ( E , . . . , E m ; F ) . Both vectorspaces are Banach spaces when endowed with the sup norm in the unit ball of B E or in product of the the unit balls B E × · · · × B E m . The following characterizations of continuous polynomials are elementary (anal-ogous results holds for multilinear operators):
Proposition 4.1.
Let
E, F be vector spaces, P ∈ P ( m E ; F ) . The following asser-tions are equivalent: (i) P ∈ P ( m E ; F ) ; (ii) P is continuous at zero; (iii) There is a constant M > such that k P ( x ) k ≤ M k x k m , for all x ∈ E ; The Polarization Formula relates polynomials and symmetric multilinear opera-tors in a very useful way. Its proof is a kind of consequence of the Leibnitz formulaand some combinatorial tricks (see [30, 45]).
Theorem 4.2 (Polarization Formula) . Let
E, F be linear spaces. If T ∈ L ( m E ; F ) is symmetric then T ( x , . . . , x m ) = 1 m !2 m X ε i = ± ε · · · ε m T ( x + ε x + · · · + ε m x m ) m , for all x , x , x , . . . , x m ∈ E. The following result is an immediate consequence of the Polarization Formula:
Corollary 4.3.
For each m -homogeneous polynomial there is a unique m -linearoperator associated to it. In other words, if P is a m -homogeneous polynomial,then there exists only one symmetric m -linear operator T (sometimes called polarof P ) such that P ( x ) = T ( x, . . . , x ) for all x. In general, if T is the symmetric m -linear operator associated to a m -homogeneouspolynomial P we have(4.1) k P k ≤ m m m ! k T k , where k P k = sup k z k =1 | P ( z ) | . The constant m m m ! is usually called polarizationconstant. If P is a homogeneous polynomial of degree m on K n given by P ( x , . . . , x n ) = X | α | = m a α x α , and L is the polar of P , then(4.2) L ( e α , . . . , e α n n ) = a α (cid:0) mα (cid:1) , where { e , . . . , e n } is the canonical basis of K n and e α k k stands for e k repeated α k times, the α j ’s are non negative integers with P nj =1 α j = α , and x α = x α · . . . · x α n n .4.1. Khinchine’s inequality.
The Khinchine inequality in its modern presenta-tion has its roots in [51]. Let ( ε i ) i ≥ be a sequence of independent Rademachervariables. Then, for any p ∈ [1 , R ,p such that, for anysequence ( a i ) of real numbers with finite support, ∞ X i =1 | a i | ! / ≤ A − R ,p Z [0 , m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X i =1 a i ε i ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω ! /p . For complex scalars it more useful (since it gives better constants) to use the follow-ing version of Khinchine’s inequality (called Khinchine’s inequality with Steinhausvariables): for any p ∈ [1 , C ,p such that, for any sequence( a i ) of complex numbers with finite support ∞ X i =1 | a i | ! / ≤ A − C ,p Z T ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X i =1 a i z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dz ! /p , with T ∞ denoting the infinite polycircle, i.e., T ∞ = (cid:8) z = ( z i ) i ∈ N ∈ C N : | z i | = 1 for all i ∈ N (cid:9) , and dz denoting the standard Lebesgue probability measure on T ∞ . The bestconstants A R ,p and A C ,p were obtained by Haagerup and K¨onig, respectively (see[36] and [41]). More precisely, • A R ,p = √ (cid:18) Γ ( p ) √ π (cid:19) /p if p > p ≈ . • A R ,p = 2 − p if p < p ; • A C ,p = Γ (cid:0) p +22 (cid:1) /p if p ∈ [1 , strange value p ≈ . p ∈ (1 ,
2) with Γ (cid:18) p + 12 (cid:19) = √ π . The notation A K ,p will be kept along this paper.Using Fubini’s theorem and Minkowski’s inequality (see, for instance, [25, Lemma2.2] for the real case and [46, Theorem 2.2] for the complex case), these inequalitieshave a multilinear version: for any n, m ≥
1, for any family ( a i ) i ∈ N m of real (resp.complex) numbers with finite support, X i ∈ N m | a i | ! / ≤ A − m R ,p Z [0 , m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈ N m a i ε (1) i ( ω ) . . . ε ( m ) i m ( ω m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dω · · · dω m ! /p ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 11 where ( ε (1) i ) , . . . , ( ε ( m ) i ) are sequences of independent Rademacher variables (resp. X i ∈ N m | a i | ! / ≤ A − m C ,p Z ( T ∞ ) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈ N m a i z (1) i . . . z ( m ) i m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dz (1) . . . dz ( m ) ! /p , in the complex case).4.2. Kahane–Salem–Zygmund’s inequality.
The essence of the Kahane–Salem–Zygmund inequalities, as we describe below, probably appeared for the first time in[40], but our approach follows the lines of Boas’ paper [11]. Paraphrasing Boas, theKahane–Salem–Zygmund inequalities use probabilistic methods to construct a ho-mogeneous polynomial (or multilinear operator) with a relatively small supremumnorm but relatively large majorant function. Both the multilinear and polynomialversions are needed for our goals.
Theorem 4.4 (Kahane–Salem–Zygmund’s inequality - Multilinear version, [11]) . Let m, n be positive integers. There exists a m -linear map T m,n : ℓ n ∞ × · · ·× ℓ n ∞ → K of the form T m,n ( z (1) , . . . , z ( m ) ) = n X i ,...,i m =1 ± z (1) i . . . z ( m ) i m such that k T m,n k ≤ p
32 log (6 m ) × n m +12 × √ n ! . The original version of the Kahane–Salem–Zygmund appears in the frameworkof complex scalars but it is simple to verify that the same result (with the sameconstants) holds for real scalars. The folowing result is corollary of the previous,now for polynomials, and it will also be important for our aims.
Theorem 4.5 (Kahane–Salem–Zygmund’s inequality - Polynomial version, [11]) . Let m, n be positive integers. Then there exists a m -homogeneous polynomial P : ℓ n ∞ → K of the form P m,n ( z ) = X | α | = d ± (cid:18) mα (cid:19) z α such that k P m,n k ≤ p
32 log (6 m ) × n m +12 × √ n ! . A corollary to Minkowski’s inequality.
Minkowski’s inequality is a verywell-known result that helps to prove that L p spaces are Banach spaces: it is thetriangle inequality for L p spaces. We need a somewhat well known result, which isa corollary of one of the many versions of Minkowski’s inequality, whose proof canbe found, for instance, in [35]. Proposition 4.6 (Corollary to Minkowski’s inequality) . For any < p ≤ q < ∞ and for any matrix of complex numbers ( c ij ) ∞ i,j =1 , ∞ X i =1 ∞ X j =1 | c ij | p q/p /q ≤ ∞ X j =1 ∞ X i =1 | c ij | q ! p/q /p . Recent “unexpected” applications to classical problems
The Bohnenblust–Hille inequality with subpolynomial constants.
TheRiemann hypothesis certainly motivated and inspired many prestigious mathemati-cians from the 20th century to study Dirichlet sums in a more extensive fashion(for instance, Bourgain, Enflo, or Montgomery [16,32,44]). Perhaps, for this reason,in the first decades of the 20th century Harald Bohr was merged in the study ofDirichlet series (see [13–15]). One of his main interests was to determine the widthof the maximal strips on which a Dirichlet series can converge absolutely but nonuniformly. More precisely, for a Dirichlet series P n a n n − s , Bohr defined σ a = inf ( r : X n a n n − s converges for Re ( s ) > r ) ,σ u = inf ( r : X n a n n − s converges uniformly in Re ( s ) > r + ε for every ε > ) , and T = sup { σ a − σ u } . Bohr’s question was: What is the value of T ?The Bohnenblust–Hille inequality was proved in 1931 by H.F. Bohnenblust andE. Hille and it is a crucial tool to answer Bohr’s problem: They proved that T = 1 / . When dealing with the Bohnenblust–Hille inequality it is elucidative to begin byproving Littlewood’s 4 / / / Theorem 5.1 (Littlewood’s 4 / . There is a constant L K ≥ such that N X i,j =1 | U ( e i , e j ) | ≤ L K k U k for every bilinear form U : ℓ N ∞ × ℓ N ∞ → K and every positive integer N. Moreover,the power / is optimal.Proof. Note that N X i,j =1 | U ( e i , e j ) | ≤ N X i =1 N X j =1 | U ( e i , e j ) | N X i =1 N X j =1 | U ( e i , e j ) | is a particular case of the procedure from Section 2. Now we just need to estimatethe two factors above. From the Khinchine inequality we have ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 13 N X i =1 N X j =1 | U ( e i , e j ) | ≤ √ N X i =1 1 Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 r j ( t ) U ( e i , e j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ √ Z N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( e i , N X j =1 r j ( t ) e j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ √ t ∈ [0 , N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ( e i , N X j =1 r j ( t ) e j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ k U k . By symmetry, the same is true swapping i and j . From Minkowski’s inequality wehave N X i =1 N X j =1 | U ( e i , e j ) | ≤ N X j =1 N X i =1 | U ( e i , e j ) | ! ≤ √ k U k and combining all these inequalities we obtain the result withL K = √ . To prove the optimality of the exponent 4 / T ,N : ℓ N ∞ → C be the bilinear form satisfying the multilinearKahane–Salem–Zygmund inequality (Theorem 4.4). Then N X i,j =1 | T ,N ( e i , e j ) | q q ≤ √ N and thus N q ≤ √ N . Next, letting N → ∞ we conclude that q ≥ . (cid:3) The natural generalization of Littlewood’s 4 / m > canbe replaced by mm +1 , and this exponent is optimal. More precisely, it asserts that,for any m ≥
2, there exists a constant C K ,m ≥ m -linear forms T : ℓ N ∞ × · · · × ℓ N ∞ → K ,(5.1) N X i ,...,i m =1 (cid:12)(cid:12) T ( e i , . . . , e i m ) (cid:12)(cid:12) mm +1 m +12 m ≤ C K ,m k T k , and all N .This result was overlooked and, sometimes, rediscovered during the last 80 years.Different approaches led to different values of the constants C m . Let us denote theoptimal constants satisfying equation (5.1) above by B mult K ,m . As a matter of fact,controlling the growth of the constants B mult K ,m is crucial for applications, as it isbeing left clear along the paper. Now we show how a suitable use of H¨older’s inequality (Theorem 3.2) providesa very simple proof of the Bohnenblust–Hille inequality, with (so far!) the bestknown constants.With the ingredients of Section 4 we can easily obtain an inductive formula forB mult K ,m . We present a sketch of the proof (more details can be found in [7]; we alsorefer to the excellent survey [28]). Theorem 5.2 (Bohnenblust–Hille inequality) . For any positive integer m , thereexists a constant B mult K ,m ≥ such that, for all m -linear forms L : ℓ N ∞ × · · · × ℓ N ∞ → K and all N , (5.2) N X i ,...,i m =1 (cid:12)(cid:12) L ( e i , . . . , e i m ) (cid:12)(cid:12) mm +1 m +12 m ≤ B mult K ,m k L k , with B mult K , = 1 and B mult K ,m ≤ A − K , kk +1 B mult K ,k . Proof.
We present a simple proof for the case k = m −
1, which is the most im-portant, since it provides better constants (and the proof for other values of k issimilar). The proof for R is essentially the same as the proof for C , so we presentonly the proof for the complex case. Let n ≥ L = P i ∈ N m a i z (1) i . . . z ( m ) i m be an m -linear form on ℓ N ∞ × · · · × ℓ N ∞ .From the Khinchine inequality we have X i S X i ˆ S | a i | × m − m m m − ≤ A − C , m − m B mult C ,m − k L k . with exponents ( q , . . . , q m ) = (cid:18) m − m , . . . , m − m , (cid:19) From the “Minkowski inequality” (Proposition 4.6) we can obtain analogous esti-mates if we take the 2 in the last position and move it backwards making it takeevery position from the last to the first; in other words, considering the followingexponents: (cid:18) m − m , . . . , , m − m (cid:19) , . . . , (cid:18) , m − m , . . . , m − m (cid:19) and the same constant. Using the H¨older inequality for multiple exponents we reachthe result. (cid:3) Using the values of the constants A K ,p we conclude that(5.3) B mult C ,m ≤ m Y j =2 Γ (cid:18) − j (cid:19) j − j . For real scalars and m ≥ mult R ,m ≤ − m m Y j =14 Γ (cid:16) − j (cid:17) √ π j − j ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 15 and B mult R ,m ≤ m Y j =2 j − = (cid:16) √ (cid:17) P m − j =1 /j . for 2 ≤ m ≤ a priori no clues on their behavior.The following consequences of Theorem 5.2 taken from [7] are illuminating: • There exists κ > m ≥ mult C ,m ≤ κ m − γ < κ m . . • There exists κ > m ≥ mult R ,m ≤ κ m − log 2 − γ < κ m . . It is interesting to note that some old estimates B mult K ,m can be easily recovered justby choosing different ( q , . . . , q m ) when using H¨older’s inequality (or using Theorem5.2 directly). For instance, • Davie ([22], 1973). B mult K ,m ≤ (cid:16) √ (cid:17) m − . Using the Khinchine inequality, we have n X i =1 . . . n X i m =1 | a i | q m ! qm − qm . . . q q q ≤ (cid:16) √ (cid:17) m − k L k for ( q , . . . , q m ) = (1 , , . . . , q , . . . , q m ) = (2 , , . . . , , . . . , ( q , . . . , q m ) = (2 , . . . , , mult K ,m ≤ (cid:16) √ (cid:17) m − . • Pellegrino and Seoane-Sep´ulveda ([48], 2012).B mult K ,m ≤ A − m/ K , mm +2 B mult K ,m/ for m even, andB mult K ,m ≤ (cid:18) A − m − K , m − m +1 B mult K , m − (cid:19) m − m (cid:18) A − m K , m +2 m +3 B mult K , m +12 (cid:19) m +12 m , for m odd.When m is even and k = m/
2, we use Khinchine inequality to obtain estimatesfor the inequalities with the exponent( q , . . . , q m ) = (cid:18) mm + 2 , . . . , mm + 2 , , . . . , (cid:19) and using the Minkowski inequality the same estimate is obtained for( q , . . . , q m ) = (cid:18) , . . . , , mm + 2 , . . . , mm + 2 (cid:19) . Using Proposition 5.2 we obtainB mult K ,m ≤ A − m/ K , mm +2 B mult K ,m/ . The case m odd is somewhat similar, although it needs a little trick . It is worthmentioning that these estimates from [48] can be somewhat derived from abstractresults appearing in [25].5.2. Quantum Information Theory.
Here we shall briefly describe a resultby Montanaro [43, Theorem 5] which provided an application for the optimalBohnenblust-Hille constants within the field of Quantum Physics. This presen-tation is based on Schwarting’s Ph.D. dissertation [50, Section 2.2.5]. For a moredetailed information we refer the interested reader to the Ph.D. dissertation ofBri¨et [17, Chapter 1], which provides a very clear introduction to the whole topicof nonlocal games.A classical nonlocal game is a pair G = ( A, π ) consisting on a function (called predicate ) A : A×B×S×S → {± } and a probability distribution π : S×T → [0 , referee and two players (usuallycalled Alice and Bob). When the game starts, the referee picks a question ( s, t ) ∈S × T according to the probability distribution π and, then, sends it to Alice andBob, who must reply independently (they are not allowed to communicate betweeneach other once the game has begun) by providing an answer a ∈ A and b ∈ B each one. The players win the game if A ( a, b, s, t ) = 1, and lose otherwise. Theplayers’ goal is to maximize their chance of winning. A XOR game is a nonlocalgame in which the answer sets A , B are {± } and the predicate A depends onlyon the exclusive-OR (XOR) of the answers given by the players and the value ofa Boolean function S × T → {± } , which from the predicate may be seen as amatrix with entries on {± } . A game with m -players is described similarly in thefollowing fashion.An m -player XOR (exclusive OR) game is a pair G = ( π, A ) consisting of a matrix A = ( a i ) i ∈M ( m,n ) , for which each entry a i ∈ {± } , and a probability distribution π : M ( m, n ) → [0 , the referee picking an m -tuple i =( i , . . . , i m ) ∈ M ( m, n ) according to the probability distribution π and sendingeach question i k to the player k , which, by means of a classical strategy, must replyupon this question with a (deterministic) answer map y k : { , . . . , n } → {± } . Theplayers win if and only if the product of their answers equals the correspondingentry in the matrix A , that is if y ( i ) · · · y m ( i m ) = a i . Concerning the complexity of a XOR game, one defines the bias β ( G ) to be thegreatest difference between the chance of winning and the chance of loosing thegame for the optimal classical strategy. Therefore, the classical bias of an m -playerXOR game is given by β ( G ) = max y ,...,y m ∈{± } n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈M ( m,n ) π ( i ) a i y ( i ) · · · y m ( i m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . If we define the m -linear map T : ℓ n ∞ × · · · × ℓ n ∞ → R by T ( e i , . . . , e i m ) := a i π ( i ),then the bias will be β ( G ) = k T k . ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 17 A natural problem is to find the game for which the classical bias is minimized.It is known that there exists an m -player XOR game G for which β ( G ) ≤ n − m − (see [33]). Using the Bohnenblust-Hille inequality it is straightforward to obtainlower bounds for the classical bias of an m -player XOR games (see [43, Theorem5]). Theorem 5.3. [43, Theorem 5]
For every m -player XOR game G = ( π, A ) , β ( G ) ≥ κm . n − m , where κ > is an universal constant.Proof. Define the m -linear form T : ℓ n ∞ × · · ·× ℓ n ∞ → R by T ( e i , . . . , e i m ) := a i π ( i ).Then, X i ∈M ( m,n ) | T ( e i , . . . , e i m ) | = X i ∈M ( m,n ) π ( i ) = 1 . Applying H¨older’s inequality and the Bohnenblust-Hille, we conclude that X i ∈M ( m,n ) | T ( e i , . . . , e i m ) | ≤ X i ∈M ( m,n ) | T ( e i , . . . , e i m ) | mm +1 m +12 m X i ∈M ( m,n ) m − m ≤ B mult R ,m n m − k T k = B mult R ,m n m − β ( G ) . Using the best known estimates for the multilinear Bohnenblust–Hille inequalitywe conclude that β ( G ) ≥ κm − log 2 − γ n m − > κm . n − m . (cid:3) This result, according to Montanaro (see [43, p.4]), implies a very particularcase of a conjecture of Aaronson and Ambainis (see [1]). Also, recent advanceson the real polynomial Bohnenblust-Hille inequality (see, e.g., [18, 31]), combinedwith the CHSH inequality (due to Clauser, Horne, Shimony, and Holt in the late1960’s), can be employed in the proof of Bell’s theorem, which states that certainconsequences of entanglement in quantum mechanics cannot be reproduced by localhidden variable theories. We refer the interested reader to the seminal paper, [21],in which more informtaion regarding this CHSH inequality can be found.5.3.
Power series and the Bohr radius problem .
The following question wasaddressed by H. Bohr in 1914:
How large can the sum of the mudulii of the terms of a convergentpower series be?
The answer was given by the following theorem, which was independently obtainby Bohr, Riesz, Schur, and Wiener:
Theorem 5.4.
Suppose that a power series P ∞ k =0 c k z k converges for z in the unitdisk, and (cid:12)(cid:12)P ∞ k =0 c k z k (cid:12)(cid:12) < when | z | < . Then P ∞ k =0 (cid:12)(cid:12) c k z k (cid:12)(cid:12) < when | z | < / . Moreover, the radius / is the best possible. Following Boas and Khavinson [10], the Bohr radius K n of the n -dimensionalpolydisk is the largest positive number r such that all polynomials P α a α z α on C n satisfy sup z ∈ r D n X α | a α z α | ≤ sup z ∈ D n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X α a α z α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The Bohr radius K was estimated by H. Bohr, M. Riesz, I. Schur and F. Wiener,and it was shown that K = 1 / n ≥
2, exact values of K n areunknown. In [10], it was proved that(5.5) 13 r n ≤ K n ≤ r log nn . The paper by Boas and Khavinson, [10], motivated many other works, connectingthe asymptotic behavior of K n to various problems in Functional Analysis (geome-try of Banach spaces, unconditional basis constant of spaces of polynomials, etc.);we refer to [26] for a panorama of the subject. Hence there was a big motivationin recent years in determining the behavior of K n for large values of n .In [23], the lefthand side inequality of (5.5) was improved toK n ≥ c p log n/ ( n log log n ) . In [24], using the hypercontractivity of the polynomial Bohnenblust–Hille inequality,the authors showed that(5.6) K n = b n r log nn with 1 √ o (1) ≤ b n ≤ . In this section we sketch how the H¨older inequality for mixed sums played a fun-damental role in the final answer to the solution, given in [7], to the Bohr radiusproblem: lim n →∞ K n q log nn = 1 . The solution has several ingredients, including the polynomial Bohnenblust–Hilleinequality. Using (4.1), Bohnenblust and Hille were also able to have a polynomialversion of this inequality: for any m ≥
1, there exists a constant D m ≥ m -homogeneous polynomial P ( z ) = P | α | = m a α z α on c , X | α | = m | a α | mm +1 m +12 m ≤ D m k P k , with D m = (cid:16) √ (cid:17) m − m m ( m + 1) m +12 m ( m !) m +12 m . In fact, it is not difficult to use polarization and obtain the polynomial Bohnenblust-Hille inequality by using the multilinear Bohnenblust-Hille inequality, but with bad constants (the following approach can be essentially found in [27, Lemma 5]). In ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 19 fact, if L is the polar of P , from (4.2) we have X | α | = m | a α | mm +1 = X | α | = m (cid:18)(cid:18) mα (cid:19) | L ( e α , . . . , e α n n ) | (cid:19) mm +1 = X | α | = m (cid:18) mα (cid:19) mm +1 | L ( e α , . . . , e α n n ) | mm +1 . However, for every choice of α , the term | L ( e α , . . . , e α n n ) | mm +1 is repeated (cid:0) mα (cid:1) times in the sum n X i ,...,i m =1 | L ( e i , . . . , e i m ) | mm +1 . Thus X | α | = m (cid:18) mα (cid:19) mm +1 | L ( e α , . . . , e α n n ) | mm +1 = n X i ,...,i m =1 (cid:18) mα (cid:19) mm +1 (cid:0) mα (cid:1) | L ( e i , . . . , e i m ) | mm +1 and, since (cid:18) mα (cid:19) ≤ m !we have X | α | = m (cid:18) mα (cid:19) mm +1 | L ( e α , . . . , e α n n ) | mm +1 ≤ ( m !) m − m +1 n X i ,...,i m =1 | L ( e i , . . . , e i m ) | mm +1 . We thus have X | α | = m | a α | mm +1 m +12 m ≤ ( m !) m − m +1 n X i ,...,i m =1 | L ( e i , . . . , e i m ) | mm +1 m +12 m = ( m !) m − m n X i ,...,i m =1 | L ( e i , . . . , e i m ) | mm +1 m +12 m ≤ ( m !) m − m B mult R ,m k L k . On the other hand, since k L k ≤ m m m ! k P k we obtain X | α | = m | a α | mm +1 m +12 m ≤ B mult R ,m ( m !) m − m m m m ! k P k = B mult R ,m m m ( m !) m +12 m k P k . Let us denote the best constant D m in this inequality by B pol C ,m . In [24] it wasproved that in fact these estimates could be essentially improved to (cid:0) √ (cid:1) m − . However using the variant of H¨older’s inequality for mixed ℓ p spaces, together withsome results from Complex Analysis (see [7] for details) and with the subpolynomial estimates of the multilinear Bohnenblust–Hille inequality (Section 5), one of themain results of [7] shows that we can go much further: Theorem 5.5.
For any ε > , there exists κ > such that, for any m ≥ , B pol C ,m ≤ κ (1 + ε ) m . As we mentioned above, in [24], using the hypercontractivity of the polynomialBohnenblust–Hille inequality, the authors showed that(5.7) K n = b n r log nn with 1 √ o (1) ≤ b n ≤ . However, although (5.7) is quite precise, there was still uncertainity in the behav-ior of the number b n . By combining classical tools of Complex Analysis (Harris’inequality [38]), Bayart’s inequality [6], Wiener’s inequality [7, Lemma 6.1], andthe Kahane–Salem–Zygmund inequality (Theorem 4.5) together with Theorem 5.5the authors, in [7], were finally able to provide the final solution to the Bohr radiusproblem: Theorem 5.6.
The asymptotic growth of the n − dimensional Bohr radius is q log nn .In other words, lim n →∞ K n q log nn = 1 . The crucial step to complete the proof was the improvement of the estimates ofthe polynomial Bohnenblust–Hille inequality that was only achieved by means ofthe H¨older inequality for mixed sums.5.4.
Hardy–Littlewood’s inequality constants.
Although H¨older’s inequalityfor mixed ℓ p spaces dates back to the 1960’s, its full importance in the subjects men-tioned throughout this paper was just very recently realized. New consequences arestill appearing (see, for instance [4, 5, 19]). The last applications of the H¨older in-equality for mixed ℓ p spaces presented here concern the Hardy–Littlewood inequal-ity and the theory of multiple summing multilinear operators. As in the case of theBohnenblust–Hille inequality (Section 5) the H¨older inequality for multiple expo-nents allows a significant improvement in the constants of the Hardy–Littlewoodinequality.Let K be R or C . Given an integer m ≥
2, the Hardy–Littlewood inequality (see[2, 37, 49]) asserts that for 2 m ≤ p ≤ ∞ there exists a constant C K m,p ≥ m –linear forms T : ℓ np × · · · × ℓ np → K and all positive integers n ,(5.8) n X j ,...,j m =1 | T ( e j , . . . , e j m ) | mpmp + p − m mp + p − m mp ≤ C K m,p k T k . Using the generalized Kahane-Salem-Zygmund inequality (see [2]) one can eas-ily verify that the exponents mpmp + p − m are optimal. When p = ∞ , using that mpmp + p − m = mm +1 , we recover the classical Bohnenblust–Hille inequality (see Theo-rem 5.2 and [12]).From [7] we know that B mult K ,m has a subpolynomial growth. On the other hand,the best known upper bounds for the constants in (5.8) were, until just recently, ¨OLDER’S INEQUALITY: SOME RECENT AND UNEXPECTED APPLICATIONS 21 (cid:0) √ (cid:1) m − (see [2, 3, 29]). Although, a suitable use of Theorem 3.2 shows that (cid:0) √ (cid:1) m − can be improved (see [5]) toC R m,p ≤ (cid:16) √ (cid:17) m ( m − p (cid:0) B mult R ,m (cid:1) p − mp for real scalars and to C C m,p ≤ (cid:18) √ π (cid:19) m ( m − p (cid:0) B mult C ,m (cid:1) p − mp for complex scalars. These estimates are substantially better than (cid:0) √ (cid:1) m − becauseB mult K ,m has a subpolynomial growth. In particular, if p > m we conclude that C K m,p has a subpolynomial growth.5.5. Separately summing operators.
H¨older’s inequality is also used to gener-alize recent results on the theory of multiple summing multilinear operators. In[25], and for m -linear operators on q -cotype Banach spaces, the authors introducedthe notion separately ( r, -summing , with 1 ≤ r ≤ q < ∞ , which means that,for any ( m − r, (cid:16) qrmq +( m − r , (cid:17) -summing. In [4] it is presented the concept of n -separabilitysumming, which stands for the m -linear operators that are multiple summing in n -coordinates, when there are m − n other coordinates fixed. Using suitable in-terpolation, the authors provide N -separability from n -separability summing, with n < N ≤ m . This result generalizes the previous one and provide more efficientexponents in some special cases. Moreover, it is also useful to provides estimatesfor the constants of some variation of Bohnenblust-Hille inequalities introduced in[46, Appendix A] and [47]. References [1] S. Aaronson and A. Ambainis,
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Departamento de Matem´atica,, Universidade Federal da Para´ıba,, 58.051-900 -Jo˜ao Pessoa, Brazil.
E-mail address : [email protected] Departamento de Matem´atica,, Universidade Federal da Para´ıba,, 58.051-900 -Jo˜ao Pessoa, Brazil.
E-mail address : [email protected] Departamento de Matem´atica,, Universidade Federal da Para´ıba,, 58.051-900 -Jo˜ao Pessoa, Brazil.
E-mail address : [email protected] and [email protected] Departamento de An´alisis Matem´atico,, Facultad de Ciencias Matem´aticas,, Plazade Ciencias 3,, Universidad Complutense de Madrid,, Madrid, 28040, Spain., and,Instituto de Ciencias Matem´aticas – ICMAT, Madrid, Spain.
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