Restricting Green potentials on metric measure spaces

Abstract

Let $(M,\rho ,\nu )$ be a locally compact separable metric measure space satisfying the doubling and reverse doubling conditions. Assume that on $(M,\rho ,\nu )$ the Green function exists and satisfies a two-sided estimate. Given a nonnegative Radon measure $\mu$ on $M$, the authors investigate restricting principles for Green–Morrey potentials on $\mu$-weak-Morrey and $\mu$-Morrey spaces. With an additional assumption of the Hölder estimate of the Green function, the authors study not only restricting properties for Green–Morrey potentials on $\mu$-Campanato spaces, but also restricting properties for Green–Hardy potentials on $\mu$-Lebesgue spaces. As applications, if there is a regular Dirichlet form on $(M,\rho ,\nu )$ which corresponds to a positive definite self-adjoint operator $L$ in $L^2\left(M\right)$, then these restricting properties can be used to derive regularity properties of the duality solutions to the equation $Lu=\mu$.