On non-isotropic Riesz potentials
Given an ordered \(n\)-tuple \(\lambda= (\lambda_1,\dots, \lambda_n)\) of nonnegative numbers and a point \(x= (x_1,\dots, x_n) \in \mathbb{R}^n\), let \(\| x\|_\lambda= (| x_1|^{1/\lambda_1}+\cdots+ | x_n|^{1/\lambda_n})^{| \lambda|/n}\), where \(| \lambda|= \lambda_1+\cdots+ \lambda_n\). If \(0< \alpha< n\) and \(f\) is a nonnegative locally integrable function on \(\mathbb{R}^n\), then a generalized potential \(\Lambda_\alpha f\) is defined on \(\mathbb{R}^n\) by \[ \Lambda_\alpha f(x)= \int_{\mathbb{R}^n}\| x-y\|_\lambda^{\alpha-n} f(y)dy. \] The paper gives a necessary and sufficient condition for \(\Lambda_\alpha f\) to be not identically \(+\infty\). It also gives a pair of integrability conditions on \(f\) which together are sufficient to ensure the continuity of \(\Lambda_\alpha f\) on \(\mathbb{R}^n\).
scientific article; zbMATH DE number 1157073