Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant

The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces and $m<p\leq 2m$ asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*} for all continuous $m$-linear forms $T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$ or $\mathbb{C}.$ The case $m=2$ recovers a classical inequality proved by Hardy and Littlewood in 1934. As a consequence of the results of the present paper we show that the same inequality is valid with $2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$. In particular, for $m<p\leq m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$

Cet article est mis à disposition selon les termes de la licence C C -3.0 F .

Introduction
Littlewood's famous 4/3 inequality [19], proved in 1930, asserts that ∞ j,k=1 T(e j , e k ) 4 3 for all continuous bilinear forms T : c 0 × c 0 → C, and the exponent 4/3 cannot be improved. In some sense this result is the starting point of the theory of multiple summing operators (for recent results on summing operators we refer to [1,8,11,15] and the references therein). From now on, for all Banach spaces E 1 , . . . , E m , F and all m-linear maps T : E 1 × · · · × E m → F, we denote T := sup T(x 1 , . . . , x m ) .
From now on we denote K = R or C and, for any function f , whenever it makes sense we formally define f (∞) = lim p→∞ f (p); moreover, we adopt a 0 = ∞ for all a > 0. In general terms we have the following m-linear inequalities: for all m-linear forms T : n p × · · · × n p → K and for all positive integers n.
• If m < p ≤ 2m, then there are constants B K m, p ≥ 1 such that for all m-linear forms T : n p × · · · × n p → K and for all positive integers n.
The exponents of all above inequalities are optimal: if replaced by smaller exponents the constants will depend on n. However, looking at the above inequalities by an anisotropic viewpoint a much richer complexity arise (see, for instance, [2,3,5,6,13,23]).
The investigation of the sharp constants in the above inequalities is more than a puzzling mathematical challenge; for applications in physics we refer to [20]. The first estimates for B K m, p had exponential growth; more precisely, for any m ≥ 1. It was just quite recently that the estimates for B K m, p were refined, see for instance [5,6,9] and references therein. It was proved in [9] that for certain constants κ 1 , κ 2 > 0, where γ is the Euler-Mascheroni constant, but, as conjectured in [24], these estimates seem to be suboptimal. For 2m(m − 1) 2 < p < ∞, among other results it was shown in [5] that we also have The best known estimates of B K m, p for the case m < p ≤ 2m are ( √ 2) m−1 (see [3,17]). Recently, Cavalcante [12] has shown that these estimates are a straightforward consequence of a kind of regularity principle proved in [4], and also investigated monotonicity properties of the optimal constants. However, the search of the optimal constants in this setting (m < p ≤ 2m) is still quite intriguing. For instance, if p = m it is easy to show that the only Hardy-Littlewood type inequality n j 1 ,..., j m =1 T(e j 1 , . . . , e j m ) happens for s = ∞ (of course, here we consider the sup norm in the left-hand-side of the inequality) and in this case it is obvious that the optimal constants are B K m,m = 1. So, we have optimal constants equal to 1 for p = m and the best known constants equal to ( √ 2) m−1 for p close to m. In this paper, among other results, we show that in fact the estimates ( √ 2) m−1 are far from being optimal: we prove that , and now, since 2 (m+1)(p−m) p −→ 1, we have a smooth connection between the estimates for p = m and p > m. From now on, for p = (p 1 , . . . , p m ) ∈ [1, ∞] m and 1 ≤ k ≤ m, let We present below the estimate obtained by Dimant and Sevilla-Peris [17] for further reference: for all m-linear forms T : n p 1 × · · · × n p m → K and all positive integers n. In particular, for all m-linear forms T : n p × · · · × n p → K and all positive integers n.
is optimal, but if one works in the anisotropic setting the result is not optimal (see, for instance, [7,23]). The main results of the present paper are the forthcoming Theorems 3.3, 3.4 and 3.5 which also improve the original constants of the bilinear Hardy-Littlewood inequalities. As a consequence of these results, when m < p ≤ m + 1, we shall prove that the optimal constants of the Hardy-Littlewood inequality are uniformly bounded by 2.

A multipurpose lemma
in which j i means that the sum runs over all indexes but j i , and the infimum is taken over all norm-one m-linear maps T : n p 1 × · · · × n p m → F. The following lemma, fundamental in the proof of our main results, is based on ideas dating back to Hardy and Littlewood (see [2,16,17,18,25]). It is our belief that it is a result of independent interest, with potential applications in the theory of multiple summing operators: . . , m, and also let λ 0 , s ≥ 1.
(1) If To prove (1), let s, λ 0 be such that (2.1) is fulfilled. Let us define Notice that λ m = η 1 . Moreover, for all j = 1, . . . , m, we have λ j−1 < λ j and where the notation above denotes the conjugate number, i.e, if a ≥ 1, then 1/a + 1/a * = 1. Let us suppose that, for k ∈ {1, . . . , m}, the inequality is true for all m-linear maps T : n p 1 × · · · × n p k−1 × n q k × · · · × n q m → F and for all i = 1, . . . , m. Let us prove that for all m-linear maps T : n p 1 × · · · × n p k × n q k+1 × · · · × n q m → F and for all i = 1, . . . , m. Consider T : n p 1 × · · · × n p k × n q k+1 × · · · × n q m → F, a m-linear map and, for each By applying the induction hypothesis to T (x) , we get, for all i = 1, . . . , m, By (2.2), using the characterization of the dual of p -type spaces, we have We continue the proof by making some other changes on the arguments borrowed from [17,18,25] to encompass our relaxed hypotheses. The main difference is that we now work in a broader scenario with q j ≤ ∞ for all j, and for this task some technical modifications are in order. Since for all positive integers N and all scalars w 1 , . . . , w N , we have and this concludes the proof of (2.3) for i = k. To prove (2.3) for i k let us initially consider k m or s > λ m = η 1 . For each i = 1, . . . , m, to simplify our notation, let us denote T(e j 1 , . . . , e j m ) From Hölder's inequality with exponents

Now, by the Hölder inequality with exponents
Let us estimate separately the two factors of this product. It follows from the case i = k that In order to estimate the first factor of the product in (2.6), we first observe that, by (2.4), we obtain By the Hölder inequality with exponents T(e j 1 , . . . , e j m ) s |x j k | s T(e j 1 , . . . , e j m ) s Plugging (2.7) and (2.8) in (2.6), we obtain It remains to consider k = m and s = λ m = η 1 . Fortunately, this case is simpler than the previous and we have q,s,λ 0 (n) T , where the inequality is due to the case i = k.
The proof of (2) is similar, except for the last step (case k = m) where one may use an argument as (2.5), but it is somewhat predictable and we omit the proof.
Remark 2.2. The case q k = ∞ for all k = 1, . . . , m in (1) is known and follows the ideas from [18,25]; our approach follows the lines of the modern presentation of [17].

Main results
We begin this section by recalling a particular instance of the Contraction Principle of [16,Theorem 12.2]. From now on r i (t) are the Rademacher functions.
The next lemma seems to be a by now standard consequence of the Contraction Principle (it was recently proved, for instance, in [22, Lemma 1]) but we present a proof here for the sake of completeness.
Proof. We will proceed by induction over m. For the case m = 1 consider l ∈ B := {1, . . . , N } and A := {l}. Thus, from Lemma 3.1, and this implies where we have used the case m = 1 and the induction hypothesis on the first and second inequalities, respectively. This concludes the proof. Now we are able to prove our first main result, providing better constants for Theorem 1.1. The main difference between the proof of our next result and the original proof of [17] is that here we use Lemma 3.2 in order to get better estimates.