Boundary behavior of generalized Poisson integrals for the half-space and the Dirichlet problem for the Schrödinger operator

Abstract:

The boundary properties are investigated for the generalized Poisson integral \[ u(X) = \int _{{\mathbb {R}^n}} {k(X,y)f(y)dy,} \] where $X$ is a point of the upper half-space $\mathbb {R}_ + ^{n + 1},\;f \in {L^{\mathbf {p}}}({\mathbb {R}^n}),\;1 \leqslant {\mathbf {p}} \leqslant \infty$ and the kernel $k$ has some special properties. Our results imply the known boundary properties of the harmonic Poisson integrals on the half-space. As an application we derive a solution of the Dirichlet problem for the operator $- \Delta + c(X),\;X \in \mathbb {R}_ + ^{n + 1}$, with boundary data $f \in {L^{\mathbf {p}}}({\mathbb {R}^n})$.