New Real-Variable Characterizations of Harmonic Functions with Mixed Morrey Traces and Applications

Abstract

A well-known result of Stein–Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. We show that a solution to the elliptic equation, defined on the upper half-space, is in the essentially-bounded-Morrey space of mixed type if and only if it can be represented by the Poisson integral of a mixed Morrey function, where a Liouville property is assumed. As applications, some new real-variable characterizations of the solution to the elliptic/parabolic equation related to the Neumann/Dirichlet problem are also considered via the gluing technology.