Let (X, d, μ) be a metric measure space satisfying the doubling condition and an L2-Poincaré inequality. Consider the nonnegative operator \(\mathcal{L}\) generalized by a Dirichlet form on X. We will show that a solution u to \((-\partial^2_t+\mathcal{L})u=0\) on X × ℝ+ satisfies an α-Carleson condition if and only if u can be represented as the Poisson integral of the operator \(\mathcal{L}\) with the trace in the generalized Morrey space L2,α(X), where α is a nonnegative function defined on a class of balls in X. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space.
