On the geometric mean operator with variable limits of integration

A new criterion for the weighted L p -L q boundedness of the Hardy operator with two variable limits of integration is obtained for 0 < q < q + 1 ≤ p < ∞. This criterion is applied to the characterization of the weighted L p -L q boundedness of the corresponding geometric mean operator for 0 < q < p < ∞. © 2008 MAIK Nauka.