Lower bounds for the constants of the Hardy–Littlewood inequalities

Abstract

Given an integer m ≥ 2, the Hardy-Littlewood inequality (for real scalars) says that for all 2m ≤ p ≤ ∞, there exists a constant C R m,p ≥ 1 such that, for all continuous m-linear forms A : N p × · · · × N p → R and all positive integers N ,   N j 1 ,...,jm=1|A(e j 1 , ..., e jm )|The limiting case p = ∞ is the well-known Bohnenblust-Hille inequality; the behavior of the constants C R m,p is an open problem. In this note we provide nontrivial lower bounds for these constants.

Optimal constants for a mixed Littlewood type inequality

“…As mentioned in [20, Theorem 1] an unified version of the above two results of Hardy and Littlewood asserts that there is a constant C p,q ≥ 1 such that [20,Theorem 1] just the complex case is considered, but for a general approach including the real case we refer to [11]; moreover the exponents are optimal). The recent years witnessed an increasing interest in the study of summability of multilinear operators (see, for instance, [10,23,24]) and in estimating constants of the multilinear and polynomial Hardy-Littlewood and related inequalities (see [2,3,4,6,14,15,26]). Perhaps the main motivations are potential applications (see, for instance, [19] for applications of the real-valued case of the estimates of the Bohnenblust-Hille inequality and [7,12] for applications of the complex-valued case).…”

The best constants in the multiple Khintchine inequality

“…Using duality and (2) we also get a similar upper bound in (1). In other words, Khintchine inequality shows that we can control the sum ∞ j=1 a j r j in any L p norm by the ℓ 2 −norm of the scalar sequence (a j ) ∞ j=1 .…”

“…Let p ∈[1,2], n ∈ N, and (y jk ) n j,k=1 an array of scalars. By (4), for any choice of signs (ε j ) n j=1 , (λ k )…”

Some applications of the regularity principle in sequence spaces

“…The investigation of the optimal constants of the Hardy-Littlewood inequalities (see [1,3,4,5,6]) is motivated by their connection with the important Bohnenblust-Hille inequality (see, for instance [9,15,18,21] and the references therein).…”