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 $0<q<q+1\le p<\infty$. This criterion is applied to the characterization of the weighted $L_p$–$L_q$ boundedness of the corresponding geometric mean operator for $0<q<p<\infty$.