Hardy Type Inequalities in Classical and Grand Lebesgue Spaces \(L_{p)}\), \(0 < p\le 1\), for Quasi-Monotone Functions

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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 Ouardani, A. , Abdelkader, S. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2. 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 Download the full text 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 + References 1. Iwaniec, T. and Sbordone, C. On the Integrability of the Jacobian Under Minimal Hypotheses, Archive for Rational Mechanics and Analysis, Springer-Verlag, 1992, vol. 119, no. 2, pp. 129-143 2. Capone, C. and Fiorenza, A. On small Lebesgue spaces, Journal of Function Spaces, 2005, vol. 3, no. 1, pp. 73-89. DOI: 10.1155/2005/192538. 3. Fiorenza, A. and Karadzhov, G. E. Grand and Small Lebesgue Spaces and Their Analogs, Zeitschrift fur Analysis und ihre Anwendungen, 2004, vol. 23, no. 4, pp. 657-681. DOI: 10.4171/ZAA/1215. 4. Samko, S. G. and Umarkhadzhiev, S. M. Local Grand Lebesgue Spaces, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 96-108. DOI: 10.46698/e4624-8934-5248-n. 5. Umarkhadzhiev, S. M. Generalization of the Notion of Grand Lebesgue Space, Russian Mathematics, 2014, vol. 58, no. 4, pp. 35-43. DOI: 10.3103/S1066369X14040057. 6. Umarkhadzhiev, S. M. One-Dimensional and Multidimensional Hardy Operators in Grand Lebesgue Spaces, Azerbaijan Journal of Mathematics, 2017, vol. 7, no. 2, pp. 132-152. 7. Bandaliev, A. R. and Safarova, K. H. On Hardy Type Inequalities in Grand Lebesgue Spaces \(L_{p)\) for \(0 < p \leq 1\), Linear and Multilinear Algebra, 2022, vol. 70, no. 21, pp. 1-14. DOI: 10.1080/03081087.2021.1944968. 8. Bergh, J., Burenkov, V. and Persson, L.-E. Best Constants in Reversed Hardy's Inequalities for Quasimonotone Functions, Acta Scientiarum Mathematicarum, Szeged, 1994, vol. 59, no. 1-2, pp. 221-239. ← Contents of issue


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