Geodesic
On the Geometric Mean Operator with Variable Limits of Integration
Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 264-288

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A new criterion for the weighted $L_p$–$L_q$ boundedness of the Hardy operator with two variable limits of integration is obtained for $0$. This criterion is applied to the characterization of the weighted $L_p$–$L_q$ boundedness of the corresponding geometric mean operator for $0$. 
@article{TRSPY_2008_260_a17,
     author = {V. D. Stepanov and E. P. Ushakova},
     title = {On the {Geometric} {Mean} {Operator} with {Variable} {Limits} of {Integration}},
     journal = {Informatics and Automation},
     pages = {264--288},
     publisher = {mathdoc},
     volume = {260},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a17/}
}
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V. D. Stepanov; E. P. Ushakova. On the Geometric Mean Operator with Variable Limits of Integration. Informatics and Automation, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 264-288. http://geodesic.mathdoc.fr/item/TRSPY_2008_260_a17/