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


Author: Alexander I. Kheifits
Journal: Proc. Amer. Math. Soc. 118 (1993), 1199-1204
MSC: Primary 31B25; Secondary 32J10
DOI: https://doi.org/10.1090/S0002-9939-1993-1146864-4
MathSciNet review: 1146864
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Abstract: The boundary properties are investigated for the generalized Poisson integral

$\displaystyle 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})$.

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DOI: https://doi.org/10.1090/S0002-9939-1993-1146864-4
Article copyright: © Copyright 1993 American Mathematical Society