Convolution maximization sequence in L 1+E(Rn) for Kernels of Lorentz space
DOI:
https://doi.org/10.56294/dm20251149Keywords:
Best constants, tight sequence, existence of extremizer, Hardy-Littlewood-Sobolev inequality, weak Lp space, convolutionAbstract
Introduction: This paper addresses the existence of amplifying convolution operators on Lebesgue spaces with kernels contained in Lorentz spaces. The analysis is rooted in the framework established by Reeve in his 1983 treatment of the Hardy-Littlewood-Sobolev inequality and is driven by the problem of determining whether convolution maximizers can be characterized when the convolution kernels lie in Lorentz spaces situated between the strong and the weak classes.
Methods: The investigation capitalizes on the prior results of G.V. Kalachev and S.Yu. Sadov by using functional analytic techniques and operator-theoretic tools. Methodological steps include the systematic examination of necessary and sufficient criteria for the existence of maximizers, the application of compactness arguments in the dual space framework, and the refinement of kernel properties through Lorentz space inequalities.
Results: The analysis establishes the existence of maximizers for convolution operators when the kernel class is contained in a slightly smaller set than weak , yet encompasses the entirety of the relevant Lorentz spaces. The abstract analytic assumptions of Theorem 2.3 are converted into explicit measurable criteria in Theorem 2.4, demonstrating that kernels selected from the identified Lorentz spaces fulfill all requisite properties for the existence of convolution maximizers.
Conclusions: The exposition achieves a systematic enlargement of convolution operator theory by admitting kernels that reside within Lorentz spaces, affording explicit existence theorems for maximizers. As a consequence, the work deepens the structural analysis of extremal functions within the harmonic analysis canon and simultaneously furnishes a robust framework for prospective inquiries regarding the deployment of Lorentz-space convolution in both pure and applied mathematics.
References
[1] L. Grafakos, Classical Fourier analysis, Springer, 2008. [2] G.V. Kalachev, S.Yu. Sadov, On maximizers of a convolution operator in spaces, Sb. Math. 210:8 (2019), 1129–1147. [3] G.V. Kalachev, S.Yu. Sadov, An existence criterion for maximizers of convolution operators in , Moscow Univ. Math. Bull., 76:4 (2021), 161–167. [4] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Ann. of Math. 118:2 (1983), 349–374. [5] E. Lieb, M. Loss, Analysis, AMS, 1997. [6] L. H¨ormander, Estimates for translation invariant operators in spaces, Acta Math. 104 (1960), 93–140. [7] L. H¨ormander, The analysis of linear partial differential operators I, Springer, 1983. [8] M. Pearson, Extremals for a class of convolution operators, Houston J. Math. 25 (1999), 43–54. [10] S. Sadov, Existence of convolution maximizers in for kernels from Lorentz spaces, arXiv:2208.08783v1 [math.FA] 18 Aug 2022. [9] V.D. Stepanov, Integral convolution operators in Lebesgue spaces, D.Sc. Dissertation, Khabarovsk, 1984. (in Russian)
[1] L. Grafakos, Classical Fourier analysis, Springer, 2008. [2] G.V. Kalachev, S.Yu. Sadov, On maximizers of a convolution operator in spaces, Sb. Math. 210:8 (2019), 1129–1147. [3] G.V. Kalachev, S.Yu. Sadov, An existence criterion for maximizers of convolution operators in , Moscow Univ. Math. Bull., 76:4 (2021), 161–167. [4] E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Ann. of Math. 118:2 (1983), 349–374. [5] E. Lieb, M. Loss, Analysis, AMS, 1997. [6] L. H¨ormander, Estimates for translation invariant operators in spaces, Acta Math. 104 (1960), 93–140. [7] L. H¨ormander, The analysis of linear partial differential operators I, Springer, 1983. [8] M. Pearson, Extremals for a class of convolution operators, Houston J. Math. 25 (1999), 43–54. [10] S. Sadov, Existence of convolution maximizers in for kernels from Lorentz spaces, arXiv:2208.08783v1 [math.FA] 18 Aug 2022. [9] V.D. Stepanov, Integral convolution operators in Lebesgue spaces, D.Sc. Dissertation, Khabarovsk, 1984. (in Russian)
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Copyright (c) 2025 Afnan Ali Hasan Al-rabiaa (Author)
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