On the differentiability of Riesz potentials of functions
The Riesz potential of order \(\alpha\) of a nonnegative measurable function f on \(R^ n\) is defined by \(R_{\alpha}f(x)=\int R_{\alpha}(x-y)f(y)dy,\) where \(R_{\alpha}(x)=| x|^{\alpha - n}\) if \(\alpha <n\) and \(R_ n(x)=\log (1/| x|)\). A function u is said to be m times totally differentiable at \(x_ 0\) if there exists a polynomial P for which \(\lim_{x\to x_ 0}| x-x_ 0|^{- m}[u(x)-P(x)]=0.\) The author proves the following Theorem: Let m be a positive integer, \(p=n/m>1\) and f be a nonnegative measurable function on \(R^ n\) such that \(R_ mf\not\equiv \infty\) and \[ \int f(y)^ p(\log (2+f(y))^{\delta} dy<\infty \text{ for some } \delta >p-1. \] Then \(R_ mf\) is m times totally differentiable almost everywhere. The proof of the theorem follows the spirit of Theorem 3 in the author's paper [Hiroshima Math. J. 11, 515-524 (1981; Zbl 0481.31004)].
scientific article; zbMATH DE number 4056012