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On relations between norms in weighted Lebesgue and weighted Hölder spaces for operators with local singularities

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Abstract

The norms in weighted Hölder spaces and in weighted Lebesgue spaces are different in their character and any direct connection between the norms of these spaces should not be expected. However, in this work, a special class of operators was found, for which we obtained an inequality that connects the norms of these operators acting on weighted Lebesgue spaces and acting on weighted Hölder spaces. A description of such operators and a relation among parameters of these spaces are given. In particular, integral operators with local endpoint singularities belong to the considered class. These results can be used in the study of operators on weighted Hölder spaces, based on known results for operators acting on weighted Lebesgue spaces.

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Acknowledgments

We would like to thank Prof. S. Grudsky and Prof. N. Vasilevski for their support and helpful discussions. We are also grateful to the anonymous referee for stimulating comments that allowed us to improve the representation.

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Authors and Affiliations

  1. Mathematical Research Center, Hidalgo State University, Carretera Pachuca-Tulancingo, Km. 4.5, C.P. 42184, Pachuca, Hidalgo, Mexico

    Anna Tarasenko

  2. Advanced Research Center on Industrial Engineering, Hidalgo State University, Carretera Pachuca-Tulancingo, Km. 4.5, C.P. 42184, Pachuca, Hidalgo, Mexico

    Oleksandr Karelin

Authors
  1. Anna Tarasenko
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  2. Oleksandr Karelin
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Corresponding author

Correspondence to Oleksandr Karelin.

Additional information

To Sergey Grudsky, professor of the Department of Mathematics, CINVESTAV, on occasion of his 60th birthday.

This work was supported by Project CB-2014-01/000000000236816/10017 CONACYT.

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Tarasenko, A., Karelin, O. On relations between norms in weighted Lebesgue and weighted Hölder spaces for operators with local singularities. Bol. Soc. Mat. Mex. 22, 503–516 (2016). https://doi.org/10.1007/s40590-016-0118-6

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  • DOI: https://doi.org/10.1007/s40590-016-0118-6

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