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.

Keywords

Orlicz Space Nondecreasing Function Lorentz Space Nonincreasing Function Hardy Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.I. Javakhishvili Tbilisi State University A. Razmadze Mathematical Institute 2TbilisiGeorgia
  2. 2.Georgian Technical UniversityTbilisiGeorgia

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