加权Lebesgue空间上Erdélyi–Kober积分与Mellin分数积分的有界性
Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces
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中文总结 AI 辅助
本文研究正实轴上加权Lebesgue空间中Erdélyi–Kober与Mellin分数积分的有界性，基于加权Hardy不等式等方法推导了相关权函数条件，揭示两类算子的联系并推广了已有结果。
中文摘要 AI 辅助
本文研究正实轴$\mathbb{R}_+=(0,\infty)$上加权Lebesgue空间中Erdélyi–Kober分数积分与Mellin分数积分的有界性性质，针对不同权函数建立了确保这些算子在加权可积空间之间有界的充分条件。本文方法主要基于加权Hardy型不等式、Hölder估计以及与算子乘法结构相关的合适变量变换。首先研究一类Erdélyi–Kober型积分算子，在权函数的适当假设下推导了加权$L^p$不等式，这些结果推广了与Hardy算子和分数积分相关的若干经典不等式。随后考虑Mellin分数积分算子，得到了其在加权Lebesgue空间中的类似有界性结果。所得估计揭示了乘法调和分析框架内Erdélyi–Kober算子与Mellin型分数积分之间的密切联系，本文结果对这些分数积分算子在加权情形下提供了统一处理，并推广了多种已知的有界性结果，特别是权函数条件刻画了算子在加权$L^p$空间上的连续性，阐明了底测度空间乘法结构所起的作用。
英文摘要
In this paper, we study the boundedness properties of Erdélyi--Kober fractional integrals and Mellin fractional integrals on weighted Lebesgue spaces over $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion _{+}=$ $\left( 0,\infty \right) $. We establish sufficient conditions on different weight functions to ensure the boundedness of these operators between weighted integrable spaces. Our approach is mainly based on weighted Hardy-type inequalities, Hölder estimates, and suitable changes of variables associated with the multiplicative structure of the operators. We first investigate a class of Erdélyi--Kober type integral operators and derive weighted $L^{p}$-inequalities under appropriate assumptions on the weights. These results extend several classical inequalities related to Hardy operators and fractional integrals. We then consider Mellin fractional integral operators and obtain analogous boundedness results in weighted Lebesgue spaces. The obtained estimates reveal a close connection between Erd% élyi--Kober operators and Mellin-type fractional integrals within the framework of multiplicative harmonic analysis. The results presented in this paper provide a unified treatment of these fractional integral operators in weighted settings and generalize various previously known boundedness results. In particular, our conditions on the weights characterize the continuity of the operators on weighted $L^{p}$-spaces and illustrate the role played by the multiplicative structure of the underlying measure space.
