About one property of the function $\|x-y\|^{2-m}$

Abstract

The kernel $ h_m(x-y)=\|x-y\|^{2-m}$ is important in the theory of subharmonic functions in the space $\mathbb{R}^m (m\ge3) $. For any $y\in\mathbb{R}^m$ we consider the kernel $h_m(x-y)$ as an element of the spaces $L_p(\gamma,\mathbb{R}^m)$. In this article we give a sufficient condition on a measure $\gamma$ the function $ h_m(x-y)\in L_p(\gamma,\mathbb{R}^m)$ to be uniformly continious in the parameter $y$ in $\mathbb{R}^m $. We give examples of measures $\gamma$, which satisfy this condition.