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Lithuanian Mathematical Journal

, Volume 53, Issue 1, pp 27–39 | Cite as

Potentials with product kernels in grand Lebesgue spaces: One-weight criteria*

  • Vakhtang KokilashviliEmail author
  • Alexander Meskhi
Article

Abstract

We study the boundedness problem for fractional integral operators with product kernels and corresponding strong fractional maximal operators in unweighted and weighted grand Lebesgue spaces. Among other statements, we prove that the one-weight inequality \( {{\left\| {{T_{\alpha }}\left( {f{w^{\alpha }}} \right)} \right\|}_{{L_w^{{q),\theta q/p}}}}}\leqslant c{{\left\| f \right\|}_{{L_w^{{p),\theta }}}}} \), where q is the Hardy–Littlewood–Sobolev exponent of p, holds for potentials with product kernels T α if and only if the weight w belongs to the Muckenhoupt class A 1+q/p′ defined with respect to n-dimensional intervals with sides parallel to the coordinate axes. We also provide a motivation of choosing θq/p as the second parameter of the target space.

Keywords

grand Lebesgue spaces strong fractional maximal operator potentials with product kernels weights boundedness one-weight inequality 

MSC

42B25 42B35 46E30 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.A. Razmadze Mathematical InstituteI. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.International Black Sea UniversityTbilisiGeorgia
  3. 3.Department of Mathematics, Faculty of Informatics and Control SystemsGeorgian Technical UniversityTbilisiGeorgia
  4. 4.Abdus Salam School of Mathematical SciencesGC UniversityLahorePakistan

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