Bayart's theory of coefficient summability on prescribed supports isolates combinatorial dimension as the geometric quantity governing the concentration of a multilinear coefficient array. We settle two sharpness questions arising from this framework. First, we determine the optimal product-summability exponent for every m≥ 2 and every prescribed combinatorial dimension d∈[1,m], thereby settling Problem 5.2 of Bayart: prod(m,d) = min{m/d, m-⌈ d⌉+1}. The answer is thus the lower envelope of two distinct obstructions: the global growth of the support inside Cartesian boxes and the concentration of the support along coordinate sections. We then complete the sharpness picture for the dimensional Hardy–Littlewood bound of Bayart. For every d∈[1,m], we construct a single infinite support Λ⊂ℕ^m of exact combinatorial dimension d on which the optimal coefficient exponent is attained simultaneously for every anisotropic parameter 𝐩=(p_1,…,p_m) satisfying ∑_j=1^m 1/p_j<1. If ∑_j=1^m 1/p_j≤ 1/2, the reciprocal optimal exponent is d+1/2d-1/d∑_j=1^m1/p_j, while if 1/2≤∑_j=1^m 1/p_j<1, it is 1-∑_j=1^m1/p_j. The same support therefore captures sharply the change of behavior at the Hilbertian threshold: below it, the critical exponent retains the prescribed dimension; above it, the dimensional dependence disappears. The proofs combine discrete Holder–Brascamp–Lieb models, dimension reduction, and sparse block selection. Rectangle growth and coordinate sections govern the product problem, while cardinality and one-coordinate fibers determine the Hardy–Littlewood realization.

The Bohnenblust-Hille inequality for multilinear forms, published in Annals of Mathematics in 1931, is now known to be part of a broad family of anisotropic inequalities that also includes, for example, the mixed Littlewood and Hardy-Littlewood inequalities. In the real scalar case, the asymptotic growth of the constants appearing in these inequalities plays a crucial role in several applications across fields such as Quantum Information Theory and Computer Science. In this paper, we classify the exponents associated with these inequalities into distinct classes via an equivalence relation, and we provide a description of the growth type of the constants in each class. In particular, we show that the constants of the anisotropic m-linear Bohnenblust-Hille inequalities over the real scalars have sub-exponential growth if, and only if, the sequence of associated exponents is equivalent to the classical exponents (m+12m, . . . , 2m m+1 )infinity m=1, thereby solving a problem proposed by the third-named author and E. Teixeira in 2018. (c) 2025 Published by Elsevier Inc.

For finite-dimensional Banach spaces E and F, let ε and π denote the injective and projective tensor norms, and set ρ(E,F):=sup_0 z∈ E⊗ Fπ(z)/ε(z). For every 1<p,q<∞ we determine the growth, as d→∞, of ρ(ℓ_p^d,ℓ_q^d): ρ(ℓ_p^d,ℓ_q^d)≍_p,qd^γ(p,q), with comparison constants depending only on p, q and the scalar field, where γ(p,q)= 3/2-max{1/p,1/q}, 1<p,q<2, 1/2+min{1/p,1/q}, 2<p,q<∞, min{1/p+1/q, 2-1/p-1/q}, otherwise. Below 2 the estimate comes from an anisotropic Hardy–Littlewood inequality together with Hadamard matrices, and the region above 2 follows by finite-dimensional duality. In the mixed region the formula is exact in every dimension.

A subset A A of a vector space X X is called α \alpha -lineable whenever A A contains, except for the null vector, a subspace of dimension α \alpha . If X X has a topology, then A A is α \alpha -spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of ( α , β ) (\alpha ,\beta ) -lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of ( α , β ) (\alpha ,\beta ) -lineability/spaceability and, as applications, we extend recent results of different authors.

In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists a bilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients $\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq C_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{ \frac{1}{2}-\frac{1}{p_{2} },0\right\} },n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2} -\frac{1}{p_{1}},0\right\} }\right\} \] for a certain constant $C_{p_{1},p_{2}}$ depending just on $p_{1},p_{2}$; moreover, the exponents of $n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ whenever $p_{1},p_{2}\in\left[ 2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our techniques are applied to provide new upper bounds for the constants of a combinatorial game, known as Gale--Berlekamp switching game or unbalancing lights problem. As a consequence, we improve estimates obtained by Brown and Spencer in $1971$ and by Carlson and Stolarski in $2004$.

The main motivation of this paper is the following general problem: under what (nontrivial) conditions a vector -valued m -linear operator T : 4p x x 4p -> 4q satisfies a summability property which originally holds for all scalar -valued mlinear forms T : 4p x x 4p -> C? For instance, under what conditions on T : 4p x x 4p -> 4q , the famous Bohnenblust-Hille inequality and Hardy- Littlewood inequalities for m -linear forms are lifted to T? We prove a general result for nonlinear operators which solves this problem as a very particular case. Our methods encompass Lipschitz operators, m -linear operators and nonlinear operators under mild assumptions. We show that, even in a very nonlinear environment, if the adjoint of T is almost q -summing, then T has the desired property. A straightforward application of our main result provides a generalization of a theorem of S. Kwapien, stated originally for linear operators.

We investigate regularity estimates of quasi-minima of the Alt–Caffarelli energy functional. We prove universal Hölder continuity of quasi-minima and optimal Lipchitz regularity along their free boundaries.

We propose a continuous version of the classical Gale-Berlekamp switching game. The main results of this paper concern growth estimates for the corresponding optimization problems.

A subset A of a vector space X is called α-lineable whenever A contains, except for the null vector, a subspace of dimension α. If X has a topology, then A is α-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of ( α,β) -lineability/spaceability were introduced as an attempt to shed light to more subtle issues. In this paper, among other results, we prove some general criteria for the notion of ( α,β)-spaceability and, as applications, we extend recent results of different authors.

The multilinear Hardy-Littlewood inequalities provide estimates for the sum of the coefficients of multilinear forms T ∶ ℓ p 1 n × ⋯ × ℓ p m n → R ( or  C ) when 1 / p 1 × ⋯ × 1 / p m < 1 . In this paper we investigate the critical and super-critical cases; i.e., when 1 / p 1 × ⋯ × 1 / p m ≥ 1.