Restricting Green potentials on metric measure spaces

Abstract Let be a locally compact separable metric measure space satisfying the doubling and reverse doubling conditions. Assume that on the Green function exists and satisfies a two‐sided estimate. Given a nonnegative Radon measure on , the authors investigate restricting principles for Green–Morrey potentials on ‐weak‐Morrey and ‐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 ‐Campanato spaces, but also restricting properties for Green–Hardy potentials on ‐Lebesgue spaces. As applications, if there is a regular Dirichlet form on which corresponds to a positive definite self‐adjoint operator in , then these restricting properties can be used to derive regularity properties of the duality solutions to the equation .