On the Boundedness and Compactness of a Class of Integral Operators

Let α > 0. The operator of the formT α f ( x ) = υ ( x )x α∫ 0 xf ( 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, L p → L q estimates of which with power weights are well known. This work gives L p → L q boundedness and compactness criteria for the operator T α in the case 0 < p , q < ∞, p > max(1/α, 1).