On the Carleson duality

TL;DR

This paper characterizes the pre-dual and multipliers of a function space used in boundary value problems, extending duality concepts between Carleson measures and maximal functions to broader L_p contexts.

Contribution

It identifies the pre-dual of the space X and characterizes its pointwise multipliers as Carleson-type functions, extending duality results to L_p spaces.

Findings

Pre-dual of space X is characterized.

Pointwise multipliers are identified as Carleson-type functions.

Results extend duality between Carleson measures and maximal functions to L_p spaces.

Abstract

As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L_2-Whitney averages belongs to L_2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X, and characterize the pointwise multipliers from X to L_2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L_p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.