On the Boundedness and Compactness of a Class of Integral Operators

Abstract

Let α > 0. The operator of the form T α f ( x ) = υ ( x ) x α ∫ 0 x f ( y ) d y ( x - y ) 1 - α , x > 0 , is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lqboundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).