Geodesic
Hardy--Steklov operators and Sobolev-type embedding inequalities
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 236-262

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We characterize weighted inequalities corresponding to the embedding of a class of absolutely continuous functions into a fractional-order Sobolev space. As auxiliary results of the paper, which are also of independent interest, we obtain several new types of necessary and sufficient conditions for the boundedness of the Hardy–Steklov operator (integral operator with two variable limits) in weighted Lebesgue spaces. 
@article{TM_2016_293_a16,
     author = {M. G. Nasyrova and E. P. Ushakova},
     title = {Hardy--Steklov operators and {Sobolev-type} embedding inequalities},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {236--262},
     publisher = {mathdoc},
     volume = {293},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_293_a16/}
}
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M. G. Nasyrova; E. P. Ushakova. Hardy--Steklov operators and Sobolev-type embedding inequalities. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 236-262. http://geodesic.mathdoc.fr/item/TM_2016_293_a16/