Taylor expansion of Riesz potentials
The Riesz potentials of the form \[ U_\alpha f(x)= \int_{\mathbb{R}^n} |x-y |^{\alpha-n} f(y) dy \] are considered where \(f(y)\) is supposed to fulfil the Orlicz condition with weight \(\omega\) in the form \(\int_{\mathbb{R}^n} \varphi_p(f(y)) \omega (|y|) dy< \infty\), where \(\varphi_p(r)\) and \(\omega(r)\) are positive, monotone functions on \((0,\infty)\) satisfying certain conditions, too lengthy to state here, \(p>1\), \(0<\alpha<n\), and \(f(y)\) is a measurable function on \(\mathbb{R}^n\). Under some restrictive conditions for \(\varphi_p(r)\), if \(p> 1\), \(U_\alpha f\) is continuous everywhere on \(\mathbb{R}^n\) (with possible exception of the origin). This result was obtained by the second author in [ibid. 18, No. 1, 73-94 (1988; Zbl 0705.31004)]. One of the results of the paper under review is that \(U_\alpha f\) is also continuous when \(p=n/ \alpha>1\), \(\omega(r)\equiv 1\) and the authors give the behaviour of \(U_\alpha f(x)-U_\alpha f(0)\) when \(x\to 0\). When the above restrictions of \(\varphi(r)\) do not hold, the second author showed in a recent paper [ibid. 23, No. 1, 79-153 (1993; Zbl 0782.31004)] that \(U_\alpha f\) is continuous only on \(\mathbb{R}^n \backslash E\), where \(E\) is a set containing the origin. In the present paper the authors extend this result showing that \[ \lim_{x\to 0, x\in \mathbb{R}^n\backslash E} \biggl(K\bigl(|x|\bigr)\biggr)^{-1} \bigl(U_\alpha f(x)-P(x) \bigr)=0, \] where \(K(x)\) is the weight function with \(K(0)=0\), and \(P(x)\) a polynomial. In numerous propositions, lemmas, corollaries and theorems, the authors establish some estimates of \(U_\alpha f\), study the Taylor expansion of \(U_\alpha f\), the limit of the studied potential when \(x\to 0\), \(x\in\mathbb{R}^n \backslash E\) and its radial limit. The paper contains many useful results for potential theory.
scientific article; zbMATH DE number 863654