Hardy–Littlewood inequalities for norms of positive operators on sequence spaces

Miguel Lacruz

doi:10.1016/j.laa.2012.07.044

Abstract

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Highlights

In 1934, Hardy and Littlewood [1], using powerful but technically difficult methods, extended results of Littlewood [2] and Toeplitz [4] to give lower bounds for norms of bilinear forms on sequence spaces
We prove Theorem 1 and Theorem 2 for the special case of an operator with non negative entries
A p,q = inf d r · B p,p, where the infimum is taken over all possible factorizations

Summary

Introduction

In 1934, Hardy and Littlewood [1], using powerful but technically difficult methods, extended results of Littlewood [2] and Toeplitz [4] to give lower bounds for norms of bilinear forms on sequence spaces. If 1 ≤ p < ∞, we write p for the complex vector space of all complex sequences x = (xk) such that x p: = We write c0 for the space of all null complex sequences x = (xk) with the norm x ∞ : = sup |xk|.

Objectives

Results

Conclusion