Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators

Abstract

The weight functions u ( x ) u(x) for which R α {R_\alpha } , the Riemann-Liouville fractional integral operator of order α > 0 \alpha > 0 , is bounded from L p ( u p d x ) {L^p}({u^p}\,dx) to L q ( u q d x ) {L^q}({u^q}\,dx) , 1 > p > 1 / α 1 > p > 1/\alpha , 1 / q = 1 / p − α 1/q = 1/p - \alpha , are characterized. Further, given p p , q q with 1 / q ⩾ 1 / p − α 1/q \geqslant 1/p - \alpha , the weight functions u > 0 u > 0 a.e. (resp. v > ∞ v > \infty a.e.) for which there is v > ∞ v > \infty a.e. (resp. u > 0 u > 0 a.e.) so that R α {R_\alpha } is bounded from L p ( v p d x ) {L^p}({v^p}\,dx) to L q ( u q d x ) {L^q}({u^q}\,dx) are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.