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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 3, Pages 70–82 (Mi vmj626)

One-sided integral operators with homogeneous kernels in grand Lebesgue spaces

a Complex Research Institute named after Kh. I. Ibragimov, Russian Academy of Sciences, Groznyi
b Academy of Sciences of Chechen Republic

Abstract: Sufficient conditions and necessary conditions for the kernel and the grandiser are obtained under which one-sided integral operators with homogeneous kernels are bounded in the grand Lebesgue spaces on $\mathbb{R}$ and $\mathbb{R}^n$. Two-sided estimates for grand norms of these operators are also obtained. In addition, in the case of a radial kernel, we obtain two-sided estimates for the norms of multidimensional operators in terms of spherical means and show that this result is stronger than the inequalities for norms of operators with a nonradial kernel.

Key words: one-sided integral operators, operators with homogeneous kernels, the grand Lebesgue spaces, two-sided estimates, spherical means.

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UDC: 517.982, 517.983

Citation: S. M. Umarkhadzhiev, “One-sided integral operators with homogeneous kernels in grand Lebesgue spaces”, Vladikavkaz. Mat. Zh., 19:3 (2017), 70–82

Citation in format AMSBIB
\Bibitem{Uma17} \by S.~M.~Umarkhadzhiev \paper One-sided integral operators with homogeneous kernels in grand Lebesgue spaces \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 3 \pages 70--82 \mathnet{http://mi.mathnet.ru/vmj626}