Integral operators of potential type and their boundary properties

R. K. Seifullaev

Sbornik. Mathematics, Tome 78 (1994) no. 2, pp. 313-332

Résumé

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 $w$).