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.