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Weighted criteria for the hardy transform under the B p condition in grand lebesgue spaces and some applications

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We show that the Hardy operator

$$ Hf(x) = \frac{1}{x}\mathop {\int }\limits_0^x f(t)dt $$

from \( L_{{\text{dec}},w}^{p),\theta } \) (I) to \( L_w^{p),\theta } \) (I), 0 < p < ∞, θ > 0, I = (0, 1), is bounded if and only if the weight w belongs to the well–known class Bp restricted to the interval I. This result is applied to derive a similar assertion for the Riemann–Liouville fractional integral operator and to establish criteria for the boundedness of the Hardy–Littlewood maximal operator in the weighted grand Lorentz space \( \Lambda_w^{p),\theta } \). Bibliography: 23 titles.

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Authors and Affiliations

  1. I. Javakhishvili Tbilisi State University A. Razmadze Mathematical Institute 2, University St., Tbilisi, 0186, Georgia

    A. Meskhi

  2. Georgian Technical University, 77 Kostava Street, Tbilisi, 0175, Georgia

    A. Meskhi

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  1. A. Meskhi
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Correspondence to A. Meskhi.

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Translated from Problems in Mathematical Analysis 60, September 2011, pp. 53–64.

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Meskhi, A. Weighted criteria for the hardy transform under the B p condition in grand lebesgue spaces and some applications. J Math Sci 178, 622–636 (2011). https://doi.org/10.1007/s10958-011-0574-5

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  • DOI: https://doi.org/10.1007/s10958-011-0574-5

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