Coordinatewise multiple summing operators in Banach spaces

Abstract

We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P : ∞ → p , as well as for the convergence theory of products of vector valued Dirichlet series.

Estimates for vector valued Dirichlet polynomials

“…Similarly, we know from [12,Corollary 5.9] and [14,Corollary 8.3] that for every operator v : 1 → q we have…”

Nonlinear absolutely summing operators revisited

“…For polynomials we write P as(p;q) ( n E; F ). For other approaches we mention [9,18,21,28] and references therein. The successful notion of multiple summing multilinear operators will be mentioned in the Section 4.…”

“…Several indicators from the theory of summing operators and from the theory of (multi-) ideals show that this is one of the most adequate approaches to the nonlinear theory of absolutely summing operators. For results on multiple summing multilinear operators we refer to [10,21,60,62,68,69].…”

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

“…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]):…”