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Integral operators with homogeneous kernels in grand Lebesgue spaces

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Abstract

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ℝn as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.

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Authors and Affiliations

  1. Academy of Sciences of the Chechen Republic, Groznyi, Russia

    S. M. Umarkhadzhiev

  2. Ibragimov Complex Research Institute, Russian Academy of Sciences, Groznyi, Russia

    S. M. Umarkhadzhiev

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  1. S. M. Umarkhadzhiev
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Correspondence to S. M. Umarkhadzhiev.

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Dedicated to Professor Stefan Samko on the occasion of his 75th birthday.

Original Russian Text © S. M. Umarkhadzhiev, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 5, pp. 775–788.

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Umarkhadzhiev, S.M. Integral operators with homogeneous kernels in grand Lebesgue spaces. Math Notes 102, 710–721 (2017). https://doi.org/10.1134/S0001434617110104

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  • DOI: https://doi.org/10.1134/S0001434617110104

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