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

Stepanov, V. D.

Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 3, 645-656 (1990).

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. Martín-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

MSC:

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

Keywords:

two-weight norm inequality; Riemann-Liouville integral operator; compactness; weighted Lebesgue space

Citations:

Zbl 0674.26008; Zbl 0704.42018