Square-function estimates for singular integrals and applications to partial differential equations

Abstract

The purpose of the present paper is to continue the program of study of elliptic boundary-value problems on Lipschitz domains with boundary data in quasi-Banach Besov spaces $B_s^{p,p}(\partial \Omega)$, initiated in [13]. Introducing a modified square function which is well-adapted for handling data with a fractional amount of smoothness, we establish the well-posedness of the Dirichlet and Neumann boundary problems for the Laplacian in Lipschitz domains, for a range of indices which includes values of $p$ less than $1$. An important ingredient in this regard is establishing suitable square-function estimates for a singular integral of potential type.