On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

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-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.