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Holder's inequality: some recent and unexpected applications.

1 Introduction

When Leonard James Rogers (1862-1933) and Otto Holder (1859-1937) discovered, independently, the famous inequality that (nowadays) holds Holder's name (1889, [44]), they could have never imagined that, at that precise moment, they had just started a "revolution" in Functional Analysis (we refer to [47] for a detailed and historical exposition). This tool is a fundamental inequality between integrals and an indispensable tool for the study of, among others, [L.sub.p] spaces. Let us recall the classical [L.sub.p] version of this inequality.

Theorem 1.1 (Holder's inequality, 1889). Let ([OMEGA], [SIGMA], [mu]) be a measure space and let p, q [member of] [1, [infinity]] with 1/p + 1/q = 1 (Holder's conjugates). Then, for all measurable real or complex valued functions f and g on [OMEGA],

[integral] |fg|d[mu] [less than or equal to] [([integral] [|f|.sup.p] d[mu]).sup.1/p] [([integral] [|g|.sup.q] d[mu]).sup.1/q]

If one has p, q [member of] (1, [infinity]), f [member of] [L.sub.p]([mu]), and g [member of] [L.sub.q]([mu]), then this inequality becomes an identity if and only if [|f|.sup.p] and [|g|.sup.q] are linearly dependent in [L.sub.1]([mu]). When one has p = q = 2 we recover a form of the Cauchy--Schwarz inequality (or Cauchy--Bunyakovsky--Schwarz inequality). Also, Holder's inequality is used to prove Minkowski's inequality (the triangle inequality for [L.sub.p] spaces) and to establish that [L.sub.q]([mu]) is the dual space of [L.sub.p]([mu]) for p [member of] [1, [infinity]). Of course, we are all familiar with these classical applications of Holder's inequality.

As it happens to almost every important result in mathematics, several extensions and generalizations of it have appeared along the time. In the case of Holder's inequality, this is not different. One of the extensions is the variant of Holder's inequality for mixed [L.sub.p] spaces, which appeared in 1961, in the seminal work of A. Benedek and R. Panzone [11]. Later in the 1980's, R. Blei and J. Fournier re-introduced the inequality for several applications on Lorentz spaces and also on PDEs (see [2, 13, 39]). Mixed [L.sub.p] spaces may be seen as a pure exercise of abstraction of the original notion of [L.sub.p] spaces, but as a matter of fact we shall show that the theory developed in [11] plays a crucial role in applications to quite different frameworks; it is intriguing that, although widely known (the paper [11] has more than 100 citations, mainly related to PDEs; we refer, for instance to [2, 24, 39]) it was overlooked in important fields of mathematics. This "gap" began to be filled in 2012-2013, when Holder's inequality for mixed [L.sub.p] spaces was used as an interpolation-type result and we shall show that different fields of Mathematics and even of Physics were positively influenced.

This expository paper is arranged as follows. Section 2 presents some motivation to illustrate the subject of this article. Section 3 is devoted to the aforementioned variant of Holder's inequality (Holder's inequality for mixed sums), providing a short proof. This result was only written in a proper and organized fashion in 1961 ([11]) but, as it will be left clear along this paper, at least in the topics gathered here (Functional Analysis, Complex Analysis and Quantum Information Theory) it was surely not been taken advantage of before 2012. Our approach is quite different from the one employed in [11] and we shall follow the lines of [10]. Section 4 will recall some useful inequalities that we shall need and Section 5 focuses on recent applications of Holder's inequality for mixed sums in Functional Analysis and Quantum Information Theory, culminating with the solution of a classical problem from Complex Analysis: the Bohr radius problem. Applications to the improvement of the constants of the Hardy-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 the following two inequalities at hand, for certain complex scalar matrix [([a.sub.ij]).sup.N.sub.i,j-1]:

[N.summation over (i=1)]([N.summation over (j=1)][|[a.sub.ij]|.sup.2]).sup.[1/2]] [less than or equal to] C and [N.summation over (j=1)]([N.summation over (i=1)][|[a.sub.ij]|.sup.2]).sup.[1/2]] [less than or equal to] C (2.1)

for some constant C > 0 and all positive integers N.

How can one find an optimal exponent r and a constant [C.sub.1] > 0 such that

([N.summation over (i,j-1)][|[a.sub.ij]|.sup.2]).sup.[1/2]] [less than or equal to] [C.sub.1]

for all positive integers N? Moreover, how can one obtain a good (small) constant [C.sub.1]?

This question (at least concerning the exponent r can be solved in no less than two ways: interpolation and Holder's inequality).

First note that, by using a consequence of Minkowski's inequality (see [40]), we know that

([N.summation over i-1][([N.summation over j-1]).sup.2]).sup.[1/2]] [less than or equal to] C. (2.2)

If we use Hoolder's inequality twice, one can proceed as follows:

[mathematical expression not reproducible]

On the other hand, by means of complex interpolation (see [12]) the solution is shorter; essentially we have two mixed inequalities with exponents (1,2) in equation (2.1) and (2, 1) in equation (2.2). By interpolating these exponents with [[theta].sub.1] = [[theta].sub.2] = 1/2 we obtain an exponent (4/3, 4/3) with constant C. The optimality of the exponent 4/3 can be proved using the Kahane--Salem--Zygmund inequality (Theorem 5.1).

The use of Holder's inequality as above becomes a very arduous work as it increases the number of indexes in the sums. The reader can test the case of three sums using Holder's inequality. More precisely, as a simple illustration suppose that

[N.summation over ([sigma](i) = 1)][([N.summation over ([sigma](j) = 1)][N.summation over ([sigma](k) = 1)][|[a.sub.ijk]|.sup.2]).sup.2] [less than or equal to] C

for all bijections [sigma] : {i, j, k} [right arrow] {i, j, k} and all N. How can we find an optimal exponent r and a constant [C.sub.1] such that

[([N.summation over (i,j,j=1)][|[a.sub.ijk]|.sup.r]).sup.[1/r]] [less than or equal to] [C.sub.1]

for every N?

The search for good constants dominating the respective inequalities is important for applications (see Section 5) and has an extra ingredient when we are using the interpolative approach: the main point is that different interpolations may result in the same exponent, but the constants involved differ. Thus, we must investigate what exponents we shall use to interpolate. More precisely, as we will see in Section 5, the Bohnenblust--Hille inequality for 3-linear forms asserts that there is a constant [C.sub.3] [greater than or equal to] 1 such that, for all N and all 3-linear forms T : [l.sup.N.sub.[infinity]] x [l.sup.N.sub.[infinity]] x [l.sup.N.sub.[infinity]] [right arrow] K,

[mathematical expression not reproducible]

here, as usual, K stands for the fields of real or complex numbers, [e.sub.j] denotes the canonical vector which entries are 1 at j-th position and 0 otherwise, and [parallel]T[parallel] denotes the sup norm.

However, the exponent 3/2 can be obtained by a "multiple" interpolation of exponents of inequalities of the form

[mathematical expression not reproducible]

with

([q.sub.1], [q.sub.2], [q.sub.3]) = (1, 2, 2), (2, 1, 2) and (2, 2, 1)

or

([q.sub.1], [q.sub.2], [q.sub.3]) = ([4/3], [4/3], 2), ([4/3], 2, [4/3]) and (2, [4/3], [4/3])

and the last procedure provides quite better constants. This is a simple illustration of the core of the new advances that lead to the results presented in this survey paper.

3 Holder's inequality revisited

Essentially, the simplest version of Holder's inequality asserts that if 1/p + 1/q = 1 and ([a.sub.j]) [member of] [l.sub.p], ([b.sub.j]) [member of] [l.sub.q] then ([a.sub.j][b.sub.j]) [member of] [l.sub.1]. In this section we present a variation of this result, which may have been seen as a variant of the following general Holder's inequality presented in the classical work [11] on mixed normed [L.sub.p] spaces. We shall now work with [L.sub.p] (N) = [l.sub.p], since it is the case we are interested in. Let us recall some useful multi-index notation: for a positive integer m and [empty set] = D [subset] N, we denote the set of multi-indices i = ([i.sub.1], ..., [i.sub.m]), with each [i.sub.k] [member of] D, by

M(m, D) := {i = ([i.sub.1], ..., [i.sub.m]) [member of] [N.sup.m]; [i.sub.k] [member of] D, k = 1, ..., m} = [D.sup.m].

We also denote M(m, n) := M(m, {1, 2, ..., n}). For p = ([p.sub.1], ..., [p.sub.m]) [member of] [[1,[infinity]].sup.m], and a Banach space X, let us consider the space

[mathematical expression not reproducible]

Namely, if [p.sub.1], ..., [p.sub.m] < [infinity], a vector matrix [([x.sub.i]).sub.i[member of]M(m,N)] [member of] [l.sub.p](X) if, and only if,

[mathematical expression not reproducible]

When X = K, we just write [l.sub.p] instead of [l.sub.p](K). Also, we deal with the coordinate product of two scalar matrices a = [([a.sub.i]).sub.i[member of]M(m,n)] and b = [([b.sub.i]).sub.i[member of]M(m,n)'] i.e.,

ab:= [([a.sub.i][b.sub.i]).sub.i[member of]M(m,n)],

The following result seems to be first observed by A. Benedek and R. Panzone (see [2,3,11]):

Theorem 3.1 (Holder's inequality for mixed [l.sub.p] spaces). Let m, n, N be positive integers, and r, q(1), ..., q(N) [member of] [(0, [infinity]].sup.m] be such that [1/[r.sub.j]] = [1/[q.sub.j](1)] + *** + [1/[q.sub.j](N)], j [member of] {1,2, ..., m},

and let [a.sub.k] := ([a.sup.k.sub.i]).sub.i[member of]M(m,n)], k = 1, ..., N, be scalar matrices. Then

[mathematical expression not reproducible]

Recall that, if r, q(1), ..., q(N) [member of] [(0, [infinity]).sup.m], the previous inequality means the following:

[mathematical expression not reproducible]

Using the above result we are able to recover the interpolative inequality from [4-6,10] (Theorem 3.2 below), that we can also, in some sense, call Holder's inequality for multiple exponents. We shall illustrate along the paper several applications (in different fields) of this result. Just before that, for a positive real number [theta], let us define [a.sup.[theta]] := [([a.sup.[theta].sub.i]).sub.i[member of]M(m,n)] It is straightforward to see that

[[parallel][a.sup.[theta]][parallel].sub.q/[theta]] = [[parallel][a.sup.[theta]][parallel].sup.[theta].sub.q],

where q/[theta] := ([q.sub.1]/[theta], ..., [q.sub.m]/[theta]).

Theorem 3.2 (Holder's inequality for multiple exponents-interpolative approach). Let m, n, N be positive integers and q, q(1), ..., q(N) [member of] [[1, [infinity]).sup.m] be such that ([1/[q.sub.1], ***, [1/[q.sub.m]]] belongs to the convex hull of ([1/[q.sub.1](k)], ***, [1/[q.sub.m](k)]), k = 1, ..., N. Then for all scalar matrix a = [([a.sub.i]).sub.i[member of]M(m,n)],

[mathematical expression not reproducible]

i.e.,

[mathematical expression not reproducible]

where [[theta].sub.k] are the coordinates of ([1/[q.sub.1](k)], ***, [1/[q.sub.m](k)]) on the convex hull.

Proof. For j = 1,...,m we have

[1/[q.sub.j]] = [[[theta].sub.1]/[q.sub.j](1)] + *** + [[[theta].sub.N]/[q.sub.j](N)] = [1/[q.sub.j](1)/[[theta].sub.1]] + *** + [1/[q.sub.j](N)/[[theta].sub.N]].

Since [mathematical expression not reproducible], by the Holder inequality for mixed [l.sub.p] spaces we conclude that

[mathematical expression not reproducible]

For the sake of completeness of this article, we would also like to present the following proof, which is based on interpolation.

Proof. (Interpolative Approach) We shall follow the lines of [4, Proposition 2.1]. We shall proceed by induction on N and we also employ the fact that, for any Banach space X and [theta] [member of] [0, 1],

[l.sub.r](X) = [[[l.sub.p](X), [l.sub.q](X)].sub.[THETA]],

with [1/[r.sub.i]] = [[theta]/[p.sub.i]] + [1-[theta]/[q.sub.i]], for i = 1,..., m (see [12]). If

[1/[q.sub.i]] = [[[theta].sub.1]/[q.sub.i](1)] + *** + [[[theta].sub.N]/[q.sub.i](N)],

with [[summation].sup.N.sub.k=1] [[theta].sub.k] = 1 and each [[theta].sub.k] [member of] [0, 1], then we also have

[1/[q.sub.i]] = [[[theta].sub.1]/[q.sub.i](1)] + [1 - [[theta].sub.1]/[p.sub.1]],

setting

for i = 1, ..., m and j = 2, ..., N. So [[alpha].sub.j] [member of] [0, 1] and [[summation].sup.N.sub.j=2] [[alpha].sub.j] = 1. Therefore, by the induction hypothesis, we conclude that

[mathematical expression not reproducible]

Combining the previous result with Minkowski's inequality we have a very useful inequality (see [10, Remark 2.2]):

Corollary 3.3. Let m, n be positive integers, 1 [less than or equal to] k [less than or equal to] m and 1 [less than or equal to] s [less than or equal to] q. Then for all scalar matrix ([a.sub.i]).sub.i[member of]M(m,n)],

[mathematical expression not reproducible]

where

[rho]:= [msq/kq + (m - k)s] and [P.sub.k] (m) stands for the set of subsets S [??] {1, ..., m} with card(S)= k.

The above corollary shows that Blei's inequality (see Corollary 3.4 below) is just a 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 option to obtain good estimates for the constants of the Bohnenblust-Hille and related inequalities. Just to illustrate the strength of Theorem 3.2 and Corollary 3.3, we present here quite a simple proof (see [10]) of Blei's inequality.

Corollary 3.4 (Blei's inequality - approach by Defant, Popa, and Schwarting, [30]).

Let A and B be two finite non-void index sets. Let [([a.sub.ij]).sub.(i,j) [member of] AxB] be a scalar matrix with positive entries, and denote its columns by [[alpha].sub.j] = [([a.sub.ij]).sub.i[member of]A] and its rows by [[beta].sub.i] = [([a.sub.ij]).sub.j[member of]B]. Then,for q, [s.sub.1], [s.sub.2] [greater than or equal to] 1 with q > max([s.sub.1], [s.sub.2]) we have

[mathematical expression not reproducible]

with

w: [[1, q).sup.2] [right arrow] [0, [infinity]), w(x,y):= [[q.sup.2](x + y) - 2qxy/[q.sup.2] - xy],

f: [[1, q).sup.2] [right arrow] [0, [infinity]), f(x,y):= [[q.sup.2]x - qxy/[q.sup.2](x + y) - 2qxy].

Proof. Let us consider the exponents

(q, [s.sub.2]), ([s.sub.1], q)

and

([[theta].sub.1], [[theta].sub.2]) = (f([s.sub.2], [s.sub.1]), f([s.sub.1], [s.sub.2])).

Note that (w ([s.sub.1], [s.sub.2]), w([s.sub.1], [s.sub.2])) is obtained by interpolating (q, [s.sub.2]) and ([s.sub.1], q) with [[theta].sub.1], [[theta].sub.2], respectively. Then, from Theorem 3.2, we have

[mathematical expression not reproducible]

Now, since q > [s.sub.2] we just need to use Proposition 4.6 to change the order of the last sum.

We invite the interested reader to compare the above proof with the proof presented in [30, pages 226-227], in which the classical Holder's inequality is needed several times.

4 Some useful inequalities

The main recent advances presented here are direct or indirect consequence of the improvements obtained in the polynomial and multilinear Bohnenblust-Hille inequalities, which were obtained by using the theory of mixed Lp spaces, more specifically the variant of Holder's inequality (Theorem 3.2). Three other important ingredients are also need: the Khinchine inequality (and its version for multiple sums), Kahane--Salem--Zygmund's inequality in 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 sections of this survey.

Polynomials in Banach spaces (at least for complex scalars) are of fundamental importance in the theory of Infinite Dimensional Holomorphy (see [35, 50]). In general the theory of polynomials and multilinear operators between normed spaces has its importance in different areas of Mathematics, from Number Theory, or Dirichlet series, to Functional Analysis.

In this section we recall the concepts of polynomials and multilinear operators between Banach spaces and some results that many authors would call "folklore", and that will be needed here. Let E, [E.sub.1], ..., [E.sub.m], and F be Banach spaces. A m-linear operator T : [E.sub.1] x *** x [E.sub.m] [right arrow] F is a map that is linear in each coordinate separately. When [E.sub.1] = *** = [E.sub.m] = E we say that T is symmetric if T([x.sub.[sigma](1)], ..., [x.sub.[sigma](m)]) = T([x.sub.1], ..., [x.sub.m]) for all bijections [sigma] : {1, ..., m} [right arrow] {1, ..., m}. A m-homogeneous polynomial is a map P : E [right arrow] F such that

P(x) = T(x, ..., x)

for some m-linear operator T : E x *** x E [right arrow] F. Continuity is defined in the obvious fashion. The spaces of continuous m-homogeneous polynomials from E to F are represented by P ([.sup.m]E; F) and the space of continuous multilinear operators from [E.sub.1] x *** x [E.sub.m] to F is denoted by L ([E.sub.1], ..., [E.sub.m]; F). Both vector spaces are Banach spaces when endowed with the sup norm in the unit ball [B.sub.E] or in the product of the the unit balls [mathematical expression not reproducible].

The following characterizations of continuous polynomials are elementary (analogous results hold for multilinear operators):

Proposition 4.1. Let P [member of] P ([.sup.m]E; F). The following assertions are equivalent:

(i) P [member of] P ([.sup.m]E; F);

(ii) P is continuous at zero;

(iii) There is a constant M > 0 such that [parallel]P (x)[parallel] [less than or equal to] M [[parallel]x[parallel].sup.m], for all x [member of] E.

The Polarization Formula relates polynomials and symmetric multilinear operators in a very useful way. Its proof is a kind of consequence of the Leibniz formula and some combinatorial tricks (see [35,50]).

Theorem 4.2 (Polarization Formula). If T [member of] L([.sup.m]E; F) is symmetric then

[mathematical expression not reproducible]

for all [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.m] [member of] 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-linear operator associated to it. In other words, ifP is a m-homogeneous polynomial, then there exists only one symmetric m-linear operator T (sometimes called polar of 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-homogeneous polynomial P we have

[parallel]P[parallel] [less than or equal to] [[m.sup.m]/m!][parallel]T[parallel], (4.1)

where [parallel]P[parallel] = [sup.sub.[parallel]z[parallel]=1] |P(z)|. The constant [[m.sup.m]/m!] is usually called polarization constant.

If P is a homogeneous polynomial of degree m on [K.sup.n] given by

P([x.sub.1], ...., [x.sub.n]) = [summation over (|[alpha]| = m) [a.sub.[alpha]][x.sup.[alpha]],

and L is the polar of P, then

[mathematical expression not reproducible], (4-2)

where {[e.sub.1], ..., [e.sub.n]} is the canonical basis of [K.sup.n] and [mathematical expression not reproducible] stands for [e.sub.k] repeated [[alpha].sub.k] times, the [[alpha].sub.j]'s are non negative integers with |[alpha]| := [[summation].sup.n.sub.j=1][[alpha].sub.j] = m, and [mathematical expression not reproducible].

4.1 Khinchine's inequality

The Khinchine inequality in its modern presentation has its origins in [56]. Let [([[epsilon].sub.i]).sub.i[greater than or equal to]1] be a sequence of independent Rademacher variables. For any p [member of] (0, [infinity]), there exists a constant [A.sub.R,p] such that, given any sequence ([a.sub.i]) of real numbers with finite support,

[mathematical expression not reproducible]

For complex scalars it is more useful (since it gives better constants) to use the following version of Khinchine's inequality (called Khinchine's inequality with Steinhaus variables): for any p [member of] (0, [infinity]), there exists a constant [A.sub.C,p] such that, for any sequence ([a.sub.i]) of complex numbers with finite support

[mathematical expression not reproducible]

with [T.sup.[infinity]] denoting the infinite polycircle, i.e.,

[T.sup.[infinity]] = {z = [([z.sub.i]).sub.i[member of]N] [member of] [C.sup.N] : |[z.sub.i]| = 1 for all i [member of] N},

and dz denoting the standard Lebesgue probability measure on [T.sup.[infinity]]. The best constants [A.sub.R,p] and [A.sub.C,p] were obtained by Haagerup and Konig, respectively (see [41] and [46]). More precisely,

* [A.sub.R,p] = [1/[square root of 2]] ([[GAMMA]([1-p/2])/[square root of [pi]]]) if 1.8474 [approximately equal to] [p.sub.0] [less than or equal to] p < 2;

* [A.sub.R,p] = [2.sup.[1/2] - [1/p]] if 0 < p < [p.sub.0];

* [A.sub.C,p] = [GAMMA][([p + 2/2]).sup.1/p] if p [member of] [1, 2];

* [A.sub.K,p] = 1 if p [greater than or equal to] 2 and K = R or C.

The (apparently) strange value [p.sub.0] [approximately equal to] 1.8474 is, to be precise, the unique number [p.sub.0] [member of] (1, 2) with

The notation [A.sub.K,p] will be kept along this paper.

Using Fubini's theorem and Minkowski's inequality (see, for instance, [30, Lemma 2.2] for the real case and [51, Theorem 2.2] for the complex case), these inequalities have a multilinear version: for any n, m [greater than or equal to] 1, for any family [([a.sub.i]).sub.i[member of][N.sup.m] of real (resp. complex) numbers with finite support,

[mathematical expression not reproducible]

where ([[epsilon].sup.(1).sub.i]), . . ., ([[epsilon].sup.(m).sub.i]) are sequences of independent Rademacher variables (resp.

[mathematical expression not reproducible]

in the complex case).

4.2 Kahane--Salem--Zygmund's inequality: suitable random polynomials

The essence of the Kahane--Salem--Zygmund inequalities, as we describe below, probably appeared for the first time in [45], but our approach follows the lines of Boas' paper [ 5]. Paraphrasing Boas, the Kahane--Salem--Zygmund inequalities use probabilistic methods to construct a homogeneous polynomial (or multilinear operator) with a relatively small supremum norm but relatively large majorant function. Both the multilinear and polynomial versions are needed for our goals.

Theorem 4.4 (Kahane--Salem--Zygmund's inequality - Multilinear version, [ 5]). Let m, n be ositive integers. There exists a m-linear ma [T.sub.m,n]: [l.sup.n.sub.[infinity]] x *** x [l.sup.n.sub.[infinity]] [right arrow] K of the form

[mathematical expression not reproducible]

such that

[parallel][T.sub.m,n][parallel] [less than or equal to] [square root of 32 log (6m)] x [n.sup.[m*1/2]] x [square root of m!].

The original version of the Kahane--Salem--Zygmund inequality appears in the framework of complex scalars but it is simple to verify that the same result (with the same constants) holds for real scalars. The following result is corollary of the previous, now for polynomials, and it will also be important for our purpose.

Theorem 4.5 (Kahane--Salem--Zygmund's inequality - Polynomial version, [15]). Let m, n be positive integers. Then there exists a m-homogeneous polynomial P : [l.sup.n.sub.[infinity]] [right arrow] K of the form

[mathematical expression not reproducible]

such that

[parallel][P.sub.m,n][parallel] [less than or equal to] [square root of 32log(6m)] x[n.sup.[m*1/2]] x [square root of m!].

4.3 A corollary to Minkowski's inequality

Minkowski's inequality is a very well-known result that helps to prove that [L.sub.p] spaces are Banach spaces: it is the triangle inequality for [L.sub.p] spaces. We need a somewhat well known result, which is a corollary of one of the many versions of Minkowski's inequality, whose proof can be found, for instance, in [40].

Proposition 4.6 (Corollary to Minkowski's inequality). For any 0 < p [less than or equal to] q < [infinity] and for any matrix of complex numbers ([c.sub.ij]).sup.[infinity].sub.i,j=1],

[mathematical expression not reproducible]

5 Recent "unexpected" applications to classical problems

5.1 The Bohnenblust--Hille inequality with subpolynomial constants

The Riemann hypothesis certainly motivated and inspired many prestigious mathematicians from the 20th century to study Dirichlet sums in a more extensive fashion (for instance, Bourgain, Enflo, or Montgomery [20, 37, 49]). In the first decades of the 20th century Harald Bohr was immersed in the study of Dirichlet series (see [17-19]). One of his main interests was to determine the width of the maximal strips on which a Dirichlet series can converge absolutely but non uniformly. More precisely, for a Dirichlet series D(s) := [[summation].sub.n] [a.sub.n][n.sup.-s], where [a.sub.n] are complex coefficients and s is a complex variable, Bohr defined

[[sigma].sub.a](D) := inf {r [member of] R : D(s) converges absolutely for Re(s) > r}, [[sigma].sub.a](D) =: inf {r [member of] R : D(s) converges uniformly in Re (s) > r + [epsilon] for every [epsilon] > 0}, and

T := sup {[[sigma].sub.a](D) - [[sigma].sub.u](D): D is a Dirichlet series}.

Bohr's question was: What is the value of T?

The Bohnenblust--Hille inequality, proved in 1931 by H.F. Bohnenblust and E. Hille, is a crucial tool to answer Bohr's problem: the precise value of T is 1/2.

When dealing with the Bohnenblust--Hille inequality it is elucidative to start with proving Littlewood's 4/3 inequality, a predecessor of the Bohnenblust--Hille inequality. Littlewood's 4/3 inequality was proved in 1930 to solve a problem posed by P.J. Daniell. It is worth noticing how Holder's inequality plays a fundamental role in the argument used in the proof. We include (for the sake of completeness) a proof of the optimality of the power 4/3 using the Kahane--Salem--Zygmund inequality.

Theorem 5.1 (Littlewood's 4/3 inequality). There is a constant [L.sub.L] [greater than or equal to] 1 such that

[([N.summation over (i,j=1)][|U([e.sub.i], [e.sub.j]|.sup.[4/3]]).sup.[3/4]] [less than or equal to] [L.sub.K][parallel]U[parallel] (5.1)

for every bilinear form U : [l.sup.N.sub.[infinity]] x [l.sup.N.sub.[infinity]] [right arrow] K and every positive integer N. Moreover, the power 4/3 is optimal.

Proof. Note that

[mathematical expression not reproducible]

is a particular case of the procedure from Section 2. Now we just need to estimate the two factors above. From the Khinchine inequality we have

[mathematical expression not reproducible]

By symmetry, the same is true if we swap i and j. From Minkowski's inequality we have

[([N.summation over (i=1)][([N.summation over (j=1)]|U([e.sub.i], [e.sub.j]).sup.2]).sup.[1/2]] [less than or equal to] [([N.summation over (j=1)][([N.summation over (i=1)]|U([e.sub.i], [e.sub.j]).sup.2]).sup.[1/2]] [less than or equal to] [square root of 2][parallel]U[parallel]

and combining all of these inequalities we obtain

[L.sub.K] = [square root of 2].

In order to prove the optimality of the exponent 4/3 we can use the Kahane--Salem--Zygmund inequality. Let [T.sub.2,N]: [l.sup.N.sub.[infinity]] x [l.sup.N.sub.[infinity]] [right arrow] C be the bilinear form satisfying the multilinear Kahane--Salem--Zygmund inequality (Theorem 4.4). Then, if (5.1) holds for an exponent q > 0, we have

[([N.summation over (i,j=1)]|[T.sub.2,N]([e.sub.i], [e.sub.j])).sup.[1/q]] [less than or equal to] [square root of 2][CN.sup.[3/2]]

and thus

[N.sup.[2/q]] [less than or equal to] [square root of 2][CN.sup.[3/2]]

Next, letting N [right arrow] [infinity] we conclude that q [greater than or equal to] [4/3]

The natural generalization of Littlewood's 4/3 inequality is the Bohnenblust--Hille inequality. This inequality essentially says that for m > 2 the exponent | can be replaced by [2m/m+1], and this exponent is optimal. More precisely, it asserts that, for any m [greater than or equal to] 2, there exists a constant [C.sub.K,m] [greater than or equal to] 1 such that, for all N and all m-linear forms T : [l.sup.N.sub.[infinity]] x *** x [l.sup.N.sub.[infinity]] - K,

[mathematical expression not reproducible]. (5.2)

This result was overlooked and, sometimes, rediscovered during the last 80 years. Different approaches led to different values of the constants [C.sub.K,m]. Let us denote the optimal constants satisfying equation (5.2) above by [B.sup.mult.sub.K,m]. As a matter of fact, controlling the growth of the constants [B.sup.mult.sub.K,m] is crucial for some applications, as it is being left clear along the paper (Sections 5.2 and 5.3 deal with Quantum information theory and the Bohr radius problem, respectively).

Now we show how a suitable use of Holder's inequality (Theorem 3.2) provides a very simple proof of the Bohnenblust-Hille inequality, with (so far!) the best known constants.

With the ingredients of Section 4 we can easily obtain an inductive formula for [B.sup.mult.sub.K,m]. We present a sketch of the proof (more details can be found in [10]; we also refer to the survey [33] which provides a careful and a deep analysis of the Bohnenblust--Hille inequality).

Theorem 5.2 (Bohnenblust--Hille inequality). For any positive integer m, there exists a constant [B.sup.mult.sub.K,m] [greater than or equal to] 1 such that, for all m-linear forms L : [l.sup.N.sub.[infinity]] x *** x l[l.sup.N.sub.[infinity]] [right arrow] K and all N,

[mathematical expression not reproducible] (5.3)

with [B.sup.mult.sub.K,m]= 1 and [mathematical expression not reproducible], for any 1 [less than or equal to] k [less than or equal to] m - l.

Proof. We present a simple proof for the case k = m - 1, which is the most important, since it provides better constants (and the proof for other values of k is similar). The proof for R is essentially the same as the proof for C, so we present only the proof for the complex case. Let n [greater than or equal to] 1 and let [mathematical expression not reproducible] be an m-linear form on [l.sup.N.sub.[infinity]] x *** x [l.sup.N.sub.[infinity]].

From the Khinchine inequality we have

[mathematical expression not reproducible]

for all S [subset] {1, ..., m} with card (S) = m - 1.

From the "Minkowski inequality" (Proposition 4.6) we can obtain analogous estimates if we take the 2 in the last position and move it backwards making it take every position from the last to the first; in other words, considering the following exponents:

([2m-2/m],...,2,[2m-2/m]),...,(2, [2m-2/m],...,[2m-2/m])

and the same constant. Using the Holder inequality for multiple exponents we reach the result.

Using the values of the constants [A.sub.K,p] we conclude that

[mathematical expression not reproducible] (5.4)

For real scalars and m [greater than or equal to] 14,

[mathematical expression not reproducible] (5.5)

and

[mathematical expression not reproducible]

for 2 [less than or equal to] m [less than or equal to] 13.

However, a first look at (5.4) and (5.5) gives a priori no clues on their behavior. The following consequences of Theorem 5.2 taken from [10] are worthed to be emphasized:

* There exists [[kappa].sub.1] > 0 such that, for any m [greater than or equal to] 1,

[mathematical expression not reproducible]

* There exists [[kappa].sub.2] > 0 such that, for any m [greater than or equal to] 1,

[mathematical expression not reproducible]

It is interesting to recall that some old estimates [B.sup.mult.sub.K,m] can be easily recovered just by choosing different ([q.sub.1], ..., [q.sub.m]) when using Holder's inequality (or using Theorem 5.2 directly). For instance,

* Davie ([26], 1973).

[B.sup.mult.sub.K,m] [less than or equal to] [([square root of 2]).sup.m-1].

Using the Khinchine inequality, we have

[mathematical expression not reproducible]

for

([q.sub.1], ..., [q.sub.m]) = (1, 2, ..., 2)

Using the "Minkowski inequality" (Proposition 4.6) we obtain the same estimate for

([q.sub.1], ..., [q.sub.m]) = (2, 1, ..., 2), ..., ([q.sub.1], ..., [q.sub.m]) = (2, ..., 2, 1)

with the same constant. Now, using Theorem 3.2, we conclude the proof with

[B.sup.mult.sub.K,m] [less than or equal to] [([square root of 2]).sup.m-1].

* Pellegrino and Seoane-Sepulveda ([53], 2012).

[mathematical expression not reproducible]

Whenm is evenand k = m/2, we use Khinchine inequality to obtainestimates for the inequalities with exponent

([q.sub.1], ..., [q.sub.m]) = ([2m/m+2], ..., [2m/m+2], 2, ..., 2)

and using the Minkowski inequality the same estimate is obtained for

([q.sub.1], ..., [q.sub.m]) = (2, ..., 2, [2m/m+2], ..., [2m/m+2]).

Using Proposition 5.2 we obtain

[mathematical expression not reproducible]

The case m odd is somewhat similar, although it needs a little trick. It is worth mentioning that these estimates from [53] can be somewhat derived from abstract results appearing in [30].

The Bohnenblust--Hille inequality (multilinear and polynomial) still have interesting versions in the setting of Lorentz spaces. Recall that, given 1 [less than or equal to] p < [infinity] and 1 [less than or equal to] q [less than or equal to] [infinity], the Lorentz space [l.sub.p,q](I) ([l.sub.p,q] for short) on a nonempty set I consists of all scalar sequences x = [([x.sub.i]).sub.i[member of]I] for which the expression

[mathematical expression not reproducible]

is finite. Here, for a given x = [([x.sub.i]).sub.i[member of]I] [member of] [l.sub.[infinity]](I), we denote by x* = [([x*.sub.j]).sub.j[member of]J] the non-increasing rearrangement of x defined by

[x*.sub.j] = inf{[lambda] > 0; card({i [member of] I; |[x.sub.i]| > [lambda]}) [less than or equal to] j}, j [member of] J,

where J = {1,..., n} whenever card (I) = n, and J = N whenever I is infinite.

As mentioned in [39], Littlewood's 4/3 inequality for Lorentz spaces can be deduced from a unpublished work of G. Pisier. Using Holder's inequality for mixed [l.sub.p] spaces, J. Fournier [39] (see also [13]) was able to provide a more general result: Bohnenblust--Hille's multilinear inequality for Lorentz spaces. On this environment, the result reads as follows: for every positive integer m, there is a constant C [greater than or equal to] 1 such that, for every n and every matrix a = [([a.sub.i]).sub.i[member of]M(m,n)], we have

[mathematical expression not reproducible]

Very interesting multilinear and polynomial Bohnenblust--Hille-type inequalities in Lorentz spaces with subpolynomial and subexponential constants were obtained by A. Defant and M. Mastylo in [29].

5.2 Quantum Information Theory

Here we shall briefly describe a result by Montanaro [48, Theorem 5] which provided an application for the optimal Bohnenblust--Hille constants for real scalars within the field of Quantum Physics. This presentation is based on Schwarting's Ph.D. dissertation [55, Section 2.2.5]. For a more detailed information we refer the interested reader to the Ph.D. dissertation of Briet [21, Chapter 1], which provides a clear introduction to the whole topic of nonlocal games.

A classical nonlocal game is a pair G = (A, [pi]) consisting on a function (called predicate) A : AxBxSxT [right arrow] {[+ or -]1}, where A, B, S and T are finite sets, and a probability distribution [pi] : S x T [right arrow] [0, 1]. The game involves three parties: a person called the referee and two players (usually called Alice and Bob). When the game starts, the referee picks a question (s, t) [member of] S x T according to the probability distribution [pi] and, then, sends it to Alice and Bob, who must reply independently (they are not allowed to communicate between each other once the game has begun) by providing an answer a [member of] A and b [member of] B each one. The players win the game if A(a, b, s, t) = 1, and lose otherwise. The players' goal is to maximize their chance of winning. A XOR game is a nonlocal game in which the answer sets A, B are {[+ or -]1} and the predicate A depends only on the exclusive-OR (XOR) of the answers given by the players and the value of a Boolean function S x T [right arrow] {[+ or -]1}, which from the predicate may be seen as a matrix with entries on {[+ or -]1}. A game with m-players is described similarly in the following fashion.

An m-player XOR (exclusive OR) game is a pair G = ([pi], A) consisting of a matrix A = [([a.sub.i]).sub.i[member of]M(mn)], for which each entry [a.sub.i] [member of] {[+ or -]1}, and a probability distribution [pi] : M(m, n) [right arrow] [0, 1]. The game consists on the referee picking an m-tuple i = ([i.sub.1],..., [i.sub.m]) [member of] M (m, n) according to the probability distribution [pi] and sending each question ik to the player k, which, by means of a classical strategy, must reply upon this question with a (deterministic) answer map [y.sub.k] : {1,..., n} [right arrow] {[+ or -]1}. The players win if and only if the product of their answers equals the corresponding entry in the matrix A, that is if

[y.sub.1]([i.sub.1]) *** [y.sub.m]([i.sub.m]) = [a.sub.i].

Concerning the complexity of a XOR game, one defines the bias [beta](G) to be the greatest difference between the chance of winning and the chance of loosing the game for the optimal classical strategy. Therefore, the classical bias of an m-player XOR game is given by

[mathematical expression not reproducible]

If we define the m-linear map T : [l.sup.n.sub.[infinity]] x *** x [l.sup.n.sub.[infinity]] [right arrow] R by [mathematical expression not reproducible], then the bias will be

[beta](G) = [parallel]T[parallel].

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

[mathematical expression not reproducible]

(see [38]). Using the Bohnenblust--Hille inequality it is straightforward to obtain lower bounds for the classical bias of an m-player XOR games (see [48, Theorem 5]).

Theorem 5.3. [48, Theorem 5] For every m-player XOR game G = ([pi], A),

[mathematical expression not reproducible]

where [kappa] > 0 is an universal constant.

Proof. Define the m-linear form T : [l.sup.n.sub.[infinity]] x *** x [l.sup.n.sub.[infinity]] [right arrow] R by [mathematical expression not reproducible]. Then,

[mathematical expression not reproducible]

Applying Holder's inequality and the Bohnenblust--Hille, we conclude that

[mathematical expression not reproducible]

Using the best known estimates for the multilinear Bohnenblust--Hille inequality we conclude that

[mathematical expression not reproducible]

This result, according to Montanaro (see [48, p.4]), implies a very particular case of a conjecture of Aaronson and Ambainis (see [1]). Also, recent advances on the real polynomial Bohnenblust--Hille inequality (see, e.g., [22,36]), combined with the CHSH inequality (due to Clauser, Horne, Shimony, and Holt in the late 1960's), can be employed in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden variable theories. We refer the interested reader to the seminal paper, [25], in which more information regarding this CHSH inequality can be found.

5.3 Power series and the Bohr radius problem

The following question was addressed by H. Bohr in 1914:

How large can the sum of the moduli of the terms of a convergent ower series be?

The answer was given by the following theorem, which was independently obtained by Bohr, Riesz, Schur, and Wiener:

Theorem 5.4. Suppose that a power series [[summation].sup.[infinity].sub.k-0][c.sub.k][z.sup.k] converges for z in the unit disk, and |[[summation].sup.[infinity].sub.k-0][c.sub.k][z.sup.k]| < 1 when |z| < 1. Then [[summation].sup.[infinity].sub.k-0]|[c.sub.k][z.sup.k]| < 1 when |z| < 1/3. Moreover, the radius 1/3 is the best possible.

Following Boas and Khavinson [14], the Bohr radius [K.sub.n] of the n-dimensional polydisk is the largest positive number r such that all polynomials [[summation].sub.[alpha]][a.sub.[alpha]][z.sup.[alpha]] on [C.sup.n] satisfy

[mathematical expression not reproducible]

The Bohr radius [K.sub.1] was estimated by H. Bohr, M. Riesz, I. Schur and F. Wiener, and it was shown that [K.sub.1] = 1/3 (Theorem 5.4). For n [greater than or equal to] 2, exact values of [K.sub.n] are unknown. In [14], it was proved that

[1/3][square root of [1/n]] [less than or equal to] [K.sub.n] [less than or equal to] 2[square root of [log n/n]]. (5.6)

The paper by Boas and Khavinson, [14], motivated many other works, connecting the asymptotic behavior of [K.sub.n] to various problems in Functional Analysis (geometry of Banach spaces, unconditional basis constant of spaces of polynomials, etc.); we refer to [31] for a panorama of the subject. Hence there was a big motivation in recent years in determining the behavior of [K.sub.n] for large values of n.

In [27], the left hand side inequality of (5.6) was improved to

[K.sub.n] [greater than or equal to] c[square root of log n/(n log log n)].

In [28], using the hypercontractivity of the polynomial Bohnenblust--Hille inequality, the authors showed that

[K.sub.n] = [b.sub.n][square root of [log n/n]] with [1/[square root of 2]] + o(1) [less than or equal to] [b.sub.n] [less than or equal to] 2. (5.7)

In this section we sketch how the Holder inequality for mixed sums played a fundamental role in the final answer to the solution, given in [10], to the Bohr radius problem:

[mathematical expression not reproducible]

The solution has several ingredients, including the polynomial Bohnenblust--Hille inequality. Using (4.1), Bohnenblust and Hille were also able to have a polynomial version of this inequality: for any m [greater than or equal to] 1, there exists a constant [D.sub.m] [greater than or equal to] 1 such that, for any complex m-homogeneous polynomial P(z) = [[summation].sub.|[alpha]|=m][a.sub.[alpha]][z.sup.[alpha]] on

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

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 [32, Lemma 5]). In fact, if L is the polar of P, from (4.2) we have

[mathematical expression not reproducible]

However, for every choice of [alpha], the term

[mathematical expression not reproducible]

is repeated times in the sum

[mathematical expression not reproducible]

Thus

[mathematical expression not reproducible]

and, since

[mathematical expression not reproducible]

we have

[mathematical expression not reproducible]

We thus have

[mathematical expression not reproducible]

On the other hand, since [parallel]L[parallel] [less than or equal to] [[m.sup.m]/m!] [parallel]P[parallel] we obtain

[mathematical expression not reproducible]

Let us denote the best constant [D.sub.m] in this inequality by [B.sup.pol.sub.C,m]. In [28] it was proved that in fact these estimates could be essentially improved to [([square root of 2]).sup.m-1]. However using the variant of Holder's inequality for mixed [l.sub.p] spaces, together with some results from Complex Analysis (see [10] for details) and with the sub-polynomial estimates of the multilinear Bohnenblust--Hille inequality (Section 5), one of the main results of [10] shows that we can go much further:

Theorem 5.5. For any [epsilon] > 0, there exists [kappa] > 0 such that, for any m [greater than or equal to] 1,

[B.sup.pol.sub.C,m] [less than or equal to] [kappa][(1 + [epsilon]).sup.m].

As we mentioned above, in [28], using the hypercontractivity of the polynomial Bohnenblust--Hille inequality, the authors showed that

[K.sub.n] = [b.sub.n][square root of [log n/n]] with [1/[square root of 2]] + o(1) [less than or equal to] [b.sub.n] [less than or equal to] 2. (5.8)

However, although (5.8) is quite precise, there was still uncertainty in the behavior of the number [b.sub.n]. By combining classical tools of Complex Analysis (Harris' inequality [43]), Bayart's inequality [9], Wiener's inequality [10, Lemma 6.1], and the Kahane--Salem--Zygmund inequality (Theorem 4.5) together with Theorem 5.5 the authors, in [10], were finally able to provide the final solution to the Bohr radius problem:

Theorem 5.6. The asymptotic growth of the n--dimensional Bohr radius is [square root of [log n/n]] In other words,

[mathematical expression not reproducible]

The crucial step to complete the proof was the improvement of the estimates of the polynomial Bohnenblust--Hille inequality that was only achieved by means of the Holder inequality for mixed sums.

5.4 Hardy--Littlewood's inequality constants

Although Holder's inequality for mixed [l.sub.p] spaces dates back to the 1960's, its full importance in the subjects mentioned throughout this paper was just very recently realized. New consequences are still appearing (see, for instance [6-8, 23]). The last applications of the Holder inequality for mixed [l.sub.p] spaces presented here concern the Hardy--Littlewood inequality and the theory of multiple summing multilinear operators. As in the case of the Bohnenblust--Hille inequality (Section 5) the Holder inequality for multiple exponents allows a significant improvement in the constants of the Hardy--Littlewood inequality.

Given an integer m [greater than or equal to] 2, the Hardy--Littlewood inequality (see [4, 42, 54]) asserts that for 2m [less than or equal to] p [less than or equal to] [infinity] there exists a constant [mathematical expression not reproducible] such that, for all continuous m--linear forms [mathematical expression not reproducible] and all positive integers n,

[mathematical expression not reproducible]

Using the generalized Kahane--Salem--Zygmund inequality (see [4]) one can easily verify that the exponents [2mp/mp+p-2m] are optimal. When p = [infinity], using that [2mp/mp|p-2m]=[2m/m|1], we recover the classical Bohnenblust--Hille inequality (see Theorem 5.2 and [16]).

From [10] we know that [B.sup.mult.sub.K,m] has a subpolynomial growth. On the other hand, the best known upper bounds for the constants in (5.9) were, until just recently, [([square root of 2]).sup.m-1] (see [4,5,34]). However, a suitable use of Theorem 3.2 shows that [([square root of 2]).sup.m-1] can be improved (see [8]) to

[mathematical expression not reproducible]

for real scalars and to

[mathematical expression not reproducible]

for complex scalars. These estimates are substantially better than [([square root of 2]).sup.m-1] because [B.sup.mult.sub.K,m] has a subpolynomial growth. In particular, if p > [m.sup.2] we conclude that [C.sup.K.sub.m,p] has a subpolynomial growth.

5.5 Separately summing operators

Holder's inequality is also used to generalize recent results on the theory of multiple summing multilinear operators. In [30], and for m-linear operators on q-cotype Banach spaces, the authors introduced the notion separately (r, 1)-summing, with 1 [less than or equal to] r [less than or equal to] q < [infinity], which means that, for any (m-1)-coordinates fixed, the resulting linear operator is (r, 1)-summing. Using separately summing maps, the authors concluded that the initial operator is multiple([qrn/q+(m-1)r],1)-summing. In [6] it is presented the concept of n-separability summing, which stands for the m-linear operators that are multiple summing in n-coordinates, when there are m--n other coordinates fixed. Using suitable interpolation, the authors provide N-separability from n-separability summing, with n < N [less than or equal to] m. This result generalizes the previous one and provides more efficient exponents in some special cases. Moreover, it is also useful to provide estimates for the constants of some variation of Bohnenblust--Hille inequalities introduced in [51, Appendix A] and [52].

Acknowledgement. The authors thank the anonymous referees for important suggestions that helped to improve the final version of this survey.

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Departamento de Matematica

Universidade Federal da Paraiba

58.051-900 - Joao Pessoa, Brazil.

email: ngalbqrq@gmail.com and ngalbuquerque@pq.cnpq.br

Departamento de Matematica

Universidade Estadual da Paraiba

58.429-600 - Campina Grande, Brazil

email: gdasaraujo@gmail.com

Departamento de Matematica

Universidade Federal da Paraiba

58.051-900 - Joao Pessoa, Brazil.

email: dmpellegrino@gmail.com and pellegrino@pq.cnpq.br

Departamento de Analisis Matematico

Facultad de Ciencias Matematicas

Plaza de Ciencias 3

Universidad Complutense de Madrid

Madrid, 28040, Spain.

AND

Instituto de Matematicas Interdisciplinar (IMI)

Madrid, Spain.

email: jseoane@mat.ucm.es

N. Albuquerque G. Araujo D. Pellegrino J.B. Seoane-Sepulveda (*)

(*) D. Pellegrino is supported by CNPq Grant 401735/2013-3 - PVE - Linha 2. Juan B. Seoane Sepulveda is supported by grant MTM2015-65825-P

Received by the editors in June 2016 - In revised form in September 2016.

Communicated by F. Bastin.

2010 Mathematics Subject Classification : 47A63, 30B50, 46G25, 46B70, 47H60.

Key words and phrases : Holder's inequality, random polynomials, interpolation, Bohr radius, Kahane--Salem--Zygmund's inequality, Quantum Information Theory, Hardy--Littlewood's inequality, Bohnenblust-Hille's inequality, Khinchine's inequality, absolutely summing operators.
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Title Annotation:EXPOSITORY PAPER
Author:Albuquerque, N.; Araujo, G.; Pellegrino, D.; Seoane-Sepulveda, J.B.
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Date:Apr 1, 2017
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