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A general theory of hypersurface potentials. (English) Zbl 0855.31004

The author investigates hypersurface potentials with general kernels. Particularly, he proves

$\begin{array}{cc}\hfill \underset{t\to +0}{lim}{\int }_{{ℝ}^{n-1}}\varphi \left(x\right)K\left(x,t\right)dx& =\gamma \varphi \left(0\right)+{\int }_{{ℝ}^{n-1}}\varphi \left(x\right)K\left(x,0\right)dx,\hfill \\ \hfill \underset{\begin{array}{c}x\to {x}_{0}\\ x\in {\nu }_{{x}_{0}}\end{array}}{lim}\phantom{\rule{4pt}{0ex}}{\int }_{{\Sigma }}\varphi \left(y\right)\frac{\partial }{\partial {\nu }_{{x}_{0}}}h\left(x-y\right)d{\sigma }_{y}& =\gamma \left({x}_{0}\right)\varphi \left({x}_{0}\right)+{\int }_{{\Sigma }}\varphi \left(y\right)\frac{\partial }{\partial {\nu }_{{x}_{0}}}h\left({x}_{0}-y\right)d{\sigma }_{y},\hfill \end{array}$

where $\varphi$ belongs to the Hölder’s class, Lyapunov’s manifold ${\Sigma }$ is the boundary of a bounded domain, the values $\gamma$ and $\gamma \left({x}_{0}\right)$ are defined by the kernels, ${\nu }_{{x}_{0}}$ is the inner normal at the point ${x}_{0}\in {\Sigma }$, $K\left(x,t\right)$ and $h\left(x\right)$ are kernels of special classes. Potentials of functions of the class ${L}_{p}$ and potentials of measures are considered as well.

##### MSC:
 31B25 Boundary behavior of harmonic functions (higher-dimensional) 35B15 Almost and pseudo-almost periodic solutions of PDE
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