Two-weighted estimates of Riemann-Liouville integrals. (Russian) Zbl 0705.26015

Let \(I_ r\) be the Riemann-Liouville integral operator \[ I_ rf(x)=(\Gamma (r))^{-1}\int^{x}_{0}(x-t)^{r-1} f(t)dt,\quad r\geq 1. \] The author establishes necessary and sufficient conditions for the continuity and compactness of the operator \(I_ r\) acting from a weighted Lebesgue space \(L_{p,v}(0,\infty)\) into \(L_{q,u}(0,\infty),\) \(1<p\leq q<\infty.\) Some partial results in this direction were announced in the author’s previous work [Dokl. Akad. Nauk SSSR 302, No.5, 1059-1062 (1988; Zbl 0674.26008)]. Let us mention here the paper by F. J. MartĂ­n-Reyes and E. Sawyer [Proc. Am. Math. Soc. 106, 727-733 (1989; Zbl 0704.42018)] where a little different (but equivalent) necessary and sufficient conditions for the continuity of the operator \(I_ r\) were found.
Reviewer: B.Opic


26D10 Inequalities involving derivatives and differential and integral operators
26A33 Fractional derivatives and integrals
47B38 Linear operators on function spaces (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems