Weighted Norm Inequalities of Hardy Type for a Class of Integral Operators

Necessary and sufficient conditions for the boundedness from L υ p ( R + ) to L u p ( R + ) of Volterra integral operators of the form K f ( x ) = ∫ 0 x k ( x , y ) f ( y ) d y , where k(x,y) is a non-negative kernel under suitable monotone conditions, are given. The cases 1 <p ⩽ q < ∞, 1 < q < p < ∞ and 0 < q < l < p < ∞ are considered. The results extend the well-known weighted norm Hardy inequality.