Restricting Green potentials on metric measure spaces

Let (M,ρ,ν)$(M, \rho, \nu)$ be a locally compact separable metric measure space satisfying the doubling and reverse doubling conditions. Assume that on (M,ρ,ν)$(M, \rho, \nu)$ the Green function exists and satisfies a two‐sided estimate. Given a nonnegative Radon measure μ$\mu$ on M$M$, the authors investigate restricting principles for Green–Morrey potentials on μ$\mu$‐weak‐Morrey and μ$\mu$‐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 μ$\mu$‐Campanato spaces, but also restricting properties for Green–Hardy potentials on μ$\mu$‐Lebesgue spaces. As applications, if there is a regular Dirichlet form on (M,ρ,ν)$(M, \rho, \nu)$ which corresponds to a positive definite self‐adjoint operator L$\mathcal {L}$ in L2(M)$L^2(M)$, then these restricting properties can be used to derive regularity properties of the duality solutions to the equation Lu=μ$\mathcal {L}u=\mu$.