Integral operators of potential type and their boundary properties
R. K. Seifullaev Baku State University
Abstract:
The properties of integral operators of the form
$$
(Au)(x)= \int_{\partial D}K(x,x-y)u(y)\,dy, \quad x\in D,
$$
$D$ a domain in
$\mathbb{R}^{m+1}$,
$m\ge1$, and of singular integral operators of the form
$$
(Bu)(x_0)=\int_{\partial D}K(x_0,x_0-y)u(y)\,dy, \quad x_0\in D,
$$
are studied in the particular case when
$\partial D$ lies in the hyperplane
$\mathbb{R}^m\times\{0\}$. General methods are used to obtain estimates of the modulus of continuity of the operator in terms of the continuity of the density, partical moduli of continuity of the characteristic
$f(x,\theta)=|x-y|^mK(x,x-y)$,
$\theta=(y-x)|y-x|^{-1}$, and also characteristics describing the smoothness of
$\partial D$ or its edge (it is assumed that the kernel
$~K(x,w)$ is homogeneous of degree
$(-m)$ with respect to
$w$).
UDC:
517.518.13/14
MSC: 31B25,
45H05,
45P05 Received: 20.12.1991
Fulltext:
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