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DOI: 10.46698/q9607-8404-0437-r Hardy Type Inequalities in Classical and Grand Lebesgue Spaces \(L_{p)}\), \(0 < p\le 1\), for Quasi-Monotone Functions
Abstract:
In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces \(L_{p)} (0,1)\), \(0 < p \leq 1 \). In particular, they established similar results for the Hardy operator in classical weighted Lebesgue spaces. Moreover, it is proved that the grand Lebesgue space \(L_{p) } (0,1)\) is a quasi-Banach function space. In this work, we are interested in Hardy inequalities applied to quasi-monotonic functions in classical Lebesgue spaces and grand Lebesgue spaces. We establish the boundedness of Hardy operator for quasi-monotone functions in grand Lebesgue spaces \(L_{p)}\), \(w(0,1),\) \(0 < p \leq 1\). In addition some integral inequalities for the Hardy operator are proved in classical weighted Lebesgue spaces \(L_{p,w}(0,1)\), \(0 < p < 1,\) for quasi-monotone functions. All inequalities are proved with sharp constants. Some results of Rovshan A. Bandaliev et al. are deduced as particular cases. Also other estimates are obtained in classical Lebesgue spaces for Hardy's operator and its dual.
Keywords: inequalities, quasi-monotone functions, Hardy operators, grand Lebesgue spaces, weighted Lebesgue spaces
Language: English
For citation: Ouardani, A. and Senouci, A. Hardy Type Inequalities in Classical and Grand Lebesgue Spaces \(L_{p)}\), \(0 < p \le 1\), for Quasi-Monotone Functions, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp.70-81. DOI 10.46698/q9607-8404-0437-r
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue |
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