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Let α > 0. The operator of the form Tαf(x)=υ(x)xα∫0xf(y)dy(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).

In the paper we obtain a precise characterization of Hardy type inequalities with weights for the negative indices and the indices between 0 and 1 and establish a duality between these cases.

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