Abstract

Equivalent conditions are proved for the Hardy type weighted inequality ∥W(⋅)−1σ(⋅)1p(⋅)∫0xf(t)dt∥Lp(⋅)(0,l)≤C∥ω(⋅)1p(⋅)f(⋅)∥Lp(⋅)(0,l),      f≥0\Big\Vert W(\cdot)^{-1}\sigma(\cdot)^{\frac{1}{p(\cdot)}} \int_{0}^{x} f(t)dt \Big \Vert_{L^{p(\cdot)}(0,l)} \leq C \Big \Vert \omega(\cdot)^{ \frac{1}{p(\cdot)}} f(\cdot) \Big \Vert_{L^{p(\cdot)}(0,l)}, \; \; \; f \geq 0 to be fulfilled in the norms of a Lebesgue space with variable exponent Lp(⋅)(0,l)L^{p(\cdot)}(0,l). It is assumed that the function p(.)p(.) is a monotone function.

Author information

Contact details are reproduced from the original publication and may be historical.

Farman Mamedov

Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan
and: Oil Gas Scientific Research Project Institute of SOCAR, Baku 1012, H. Zardabi, 88A, Azerbaijan

farman-m@mail.ru

Firana M. Mammadova

Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan

mamedovafira@yahoo.com

Mushviq Aliyev

Mathematics and Mechanics Institute, National Academy of Sciences, B. Vahabzade 9, Baku 1141, Azerbaijan

a.mushfiq@rambler.ru

Keywords

  • Hardy operator
  • Hardy type inequality
  • variable exponent
  • weighted inequality
  • necessary and sufficient condition

Mathematics Subject Classification

42A05, 42B25, 26D10; 35A23

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Suggested citation

F. Mamedov, F. M. Mammadova, M. Aliyev. “Boundedness Criterions for the Hardy Operator in Weighted L^{p(·)}(0,l) Space.” Journal of Convex Analysis 22 (2015), No. 2, 553–568. https://doi.org/10.68381/jca22027

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press.