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arXiv · 2608.09401

Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces

Ferit Gürbüz
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Abstract

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.

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Ferit Gürbüz. 2026-08-10. Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces. https://arxiv.org/abs/2608.09401

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