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<p>We consider a&#xa0;class of integral operators with special Legendre functions in the kernels.The&#xa0;main content is a&#xa0;survey of results on boundedness and unitarity conditions for such operators in classical Lebesgue spaces.This class includes, for example, one-dimensional Buschman–Erdélyi operators and multidimensional operators of analogous structure, including those with integration over special pyramids.We apply Mellin transform techniques involving hypergeometric Fox functions and inversion formulas for the indicated class of integral operators.In&#xa0;the second part of the article, on&#xa0;the basis of results on the unitarity of Buschman–Erdélyi operators for special parameter values, we&#xa0;consider generalizations of a&#xa0;well-known property of Hardy operators and construct additional examples related to this problem.</p>

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Boundedness and Unitarity Conditions in Lebesgue Spaces for Integral Operators with Special Legendre Functions in the Kernels

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • S. M. Sitnik,
  • O. V. Skoromnik

摘要

We consider a class of integral operators with special Legendre functions in the kernels.The main content is a survey of results on boundedness and unitarity conditions for such operators in classical Lebesgue spaces.This class includes, for example, one-dimensional Buschman–Erdélyi operators and multidimensional operators of analogous structure, including those with integration over special pyramids.We apply Mellin transform techniques involving hypergeometric Fox functions and inversion formulas for the indicated class of integral operators.In the second part of the article, on the basis of results on the unitarity of Buschman–Erdélyi operators for special parameter values, we consider generalizations of a well-known property of Hardy operators and construct additional examples related to this problem.