A new multilinear insight on Littlewood's 4/3-inequality

Abstract

We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in p -spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous p -valued polynomials on c 0 .

Convergence of Dirichlet polynomials in Banach spaces

“…Based on our results from [13] we continue the recent study of ordinary Dirichlet series from [11]. Let X be some Banach space and m ∈ N. We call a series n a n 1 n s , s ∈ C, a Dirichlet series in X if all its coefficients a n belong to X.…”

“…Using results and techniques of Bohr [5,6], Bohnenblust-Hille [4] and [11,13], the proof will be given in Sections 3, 4 and 5.…”

“…Hence it is now no surprise that cot(X) appears, since by [21, Théorème 1.1] (see also [14, page 304]) this is exactly the infimum over all r so that id X is (r, 1)-summing (then the 'best summability' we can expect from id X ); in other words BH 1 (id X ) = cot(X). From [7,Theorem 3.2] we have BH m (id X ) ≤ cot(X) for every infinite dimensional Banach space X (even equality by [13,Proposition 7]). With Proposition 6.1 this altogether yields that for infinite dimensional Banach spaces,…”

Nonlinear absolutely summing operators revisited

“…If r 1 = · · · = r n = s we just write (r; s), and when r = s we replace (r; r) by r. For n = 1 this concept also coincides with the classical notion of absolutely summing linear operators and, for this reason, we keep the usual notation π (r;s) (T ) instead of T (r;s) for the norm of T. The essence of the notion of multiple summing multilinear operators, for bilinear operators, can also be traced back to [71]. For recent results in the theory of multiple summing operators we refer to [8,22,60,68] and references therein.…”

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]):…”