Sharp generalizations of the multilinear Bohnenblust–Hille inequality

Abstract

Abstract. We prove that the multilinear Bohnenblust-Hille is a particular case of a quite general family of optimal inequalities.

Optimal Hardy–Littlewood type inequalities for m-linear forms on $${\ell_{p}}$$ ℓ p spaces with $${1\leq p\leq m}$$ 1 ≤ p ≤ m

“…In this section we obtain partial answers for the cases not covered by our main theorem, i.e., the cases (r, p) ∈ [1, 2] × [1,2) and (r, p) ∈ (2, ∞) × [1, m]. for all m-linear forms T : n p × · · · × n p → K and all positive integers n. Moreover the optimal exponent of n is not smaller than (2m − r)/2r.…”

“…For references we mention, for instance, [2,6,13,14,21] and the very interesting survey [15]. The optimal values of B K,m are unknown; the best known upper and lower estimates for the constants in (1.1) are (see [6,18]):…”

Multiple summing maps: Coordinatewise summability, inclusion theorems and p-Sidon sets

“…Hence the assumptions of Theorem 2.1 are satisfied and this leads to (1). To prove (2), we suppose first that r k < q for all k. Let J be a maximal set of {1, .…”

Bohnenblust-Hille inequalities: analytical and computational aspects

“…Further generalizations to the anisotropic settings were obtained in Albuquerque et al (2014) and the best known estimates for C K,m,p can be found in Pellegrino (2014, 2017) and Cavalcante et al (2016). The case m < p < 2m was recently explored in Dimant and Sevilla-Peris (2016) and the constants involved were further explored in Albuquerque et al (2017), Nunes (2017), among others.…”