Uniqueness and normality for M.~Riesz potentials and solutions of the Darboux equation

A. I. Sergeyev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 141-147

Résumé

M. Riesz potentials $U_\alpha^\mu(x)=\int_{\partial\Omega}\frac{d\mu(y)}{|x-y|^{n-1+\alpha}}$ are considered where $\Omega$ is a domain in $\mathbb R^{n+1}$ with a nice boundary $\partial\Omega$, $\mu$ a Borel charge on $\partial\Omega$. These potentials satisfy the Darboux equation \begin{equation} \Delta U+\frac\alpha yU_y=0,\qquad x=(\overline x,y),\quad\overline x\in\mathbb R^n. \end{equation} Theorems of the following kind are stated: if $U^\mu_\alpha$ and $\mu$ decrease rapidly near a point $p\in\partial\Omega$ along $\partial\Omega$, then $\mu\equiv0$; analogous results are stated for solutions of (1). These results are closely connected with “normality properties”, i.e., the uniform boundedness (on compact subsets of $\Omega$) of potentials (respectively, solutions of (1)) $U^\mu_\alpha$ satisfying some growth restrictions along $\partial\Omega$. Bibl. 10 titles.