In this paper, we establish weighted norm inequalities for the harmonic function in the upper half-space, and corresponding reverse inequalities are also obtained. These weighted Morrey-Lorentz estimates generalize the \(L^p\) -isometry of the Poisson semigroup in Stein-Weiss’s book [Introduction to Fourier analysis on Euclidean spaces, 1971]. The novelty of our results lies in the weight constructed from a product of two weights \(\mu \) and \(\omega \) . The former weight \(\mu \) arises from the structure of the elliptic operator, while the latter \(\omega \) is in the Muckenhoupt class with respect to \(\mu \) . This is totally different from the two-weight theory. As applications, weighted norm inequalities and their reversed for the harmonic/caloric function related to the Neumann/Dirichlet problem are considered.
