On the Carleson duality

Abstract

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L₂ Whitney averages belongs to L₂ 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 ${\mathcal{X}}$, and characterize the pointwise multipliers from ${\mathcal{X}}$ to L₂ 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 ${\mathcal{X}}$. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.