On the Dirichlet problem for the Schrödinger equation in the upper half-space

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. Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed.