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<p>Classical Hardy’s inequalities focus on the Hardy operator and its adjoint, the Bellman operator, while fractional Hausdorff operators naturally generalize fractional versions of these two operators. In this paper, we give the necessary conditions and sufficient conditions on weight functions for the boundedness of the fractional Hausdorff operators on weighted Lebesgue spaces. Note that in the main results at least one of the weights is monotone. In particular, we establish necessary and sufficient conditions on monotone weight functions for the boundedness of the fractional Hausdorff operators on weighted Lebesgue spaces. The weights which may fit into the requirement of our results are given.</p>

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On Weighted \(\left( L_p,\,L_q\right) \)-Boundedness Characterization of Fractional Hausdorff Operator

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Rovshan A. Bandaliyev,
  • Konul A. Babayeva,
  • Khalida S. Nasirova

摘要

Classical Hardy’s inequalities focus on the Hardy operator and its adjoint, the Bellman operator, while fractional Hausdorff operators naturally generalize fractional versions of these two operators. In this paper, we give the necessary conditions and sufficient conditions on weight functions for the boundedness of the fractional Hausdorff operators on weighted Lebesgue spaces. Note that in the main results at least one of the weights is monotone. In particular, we establish necessary and sufficient conditions on monotone weight functions for the boundedness of the fractional Hausdorff operators on weighted Lebesgue spaces. The weights which may fit into the requirement of our results are given.