Two-weighted estimates of Riemann--Liouville integrals

V. D. Stepanov

Izvestiya. Mathematics , Tome 36 (1991) no. 3, pp. 669-681

Résumé

Weighted estimates \begin{equation} \left(\int\limits_0^\infty|I_rf(x)u(x)|^q\,dx\right)^{1/q}\leqslant C\left(\int\limits_0^\infty|f(x)v(x)|^p\,dx\right)^{1/p} \end{equation} are considered, where the constant $C$ does not depend on $f$, for fractional Riemann– Liouville integrals $$ I_r(f(x)=\frac {1}{\Gamma (r)}\int\limits_0^x(x-t)^{r-1}f(t)\,dt,\quad r>0, $$ and the following problem is examined: find necessary and sufficient conditions on weight functions $u$ and $v$ under which estimate (1) is valid for all functions for which the right-hand side of (1) is finite. The problem is solved for $1\leqslant p\leqslant q\leqslant\infty$ and $r>1$. This result is definitive, and it generalizes known results for integral operators when $r=1$.