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Two-Weight Weak Type Inequality for Modified Fractional Maximal Function Defined with Respect to Non-Doubling Measure and Some Applications in Non-standard Lebesgue Spaces

  • Sambourou MassinankeEmail author
Article
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Abstract

Let \( \mu \) a nonnegative Radon measure on \( {\mathbb{R}}^{d} \); \( p,q,\gamma ,k \) real numbers; \( M_{\mu ,k}^{\gamma } \) a fractional maximal operator; \( A_{p,q}^{\gamma ,k} \left( \mu \right) \) a Muckenhoupt condition associated to \( \mu \); \( L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) \) and \( F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) \) two generalized Lebesgue spaces. The purpose of the present work is double:
  • First, a characterization of the Muckenhoupt \( A_{p,q}^{\gamma ,k} \left( \mu \right) \) is given.

  • At the end, the fractional maximal \( M_{\mu ,k}^{\gamma } \) is applied on the two generalized Lebesgue spaces \( L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) \) and \( F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) \).

Keywords

Fractional maximal operator \( A_{p} \)-weights theory Generalization of Lebesgue spaces Besicovitch covering lemma Radon measure 

Notes

Acknowledgements

I thank very sincerely the reviewer for his comments which helped us to improve the presentation of our results.

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Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Department of Studies and Researches of Mathematics and Computer Sciences, Faculty of Sciences and Techniques (FST) of BamakoUniversity of BamakoBamakoMali

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