Sufficient conditions for the Lebesgue integrability of Mellin transforms

Abstract

This work is devoted to the study of the integrability of Mellin transforms under some sufficient and necessary conditions in the space $X_c^p$ of Lebesgue measurable functions f on ${\mathbb{R}}_{+}=(0,+\mathrm{\infty })$ such that ${\int }_0^{+\mathrm{\infty }}|u^{c}f(u){|}^p\frac{\mathrm{d}u}{u}<\mathrm{\infty },c\in \mathbb{R}(1<p\le 2).$ These results are inspired by a Móricz-type result in classical Fourier analysis, which generalizes a well-known theorem of Titchmarsh.

Keywords:

• Mellin transforms
• Mellin translation operator
• Lipschitz conditions

Subject Classifications:

• 42B35
• 44A05

Acknowledgments

The author would like to express his thanks to the referees. They carefully read the original manuscript and made many useful comments to improve the paper.

Availability of data and materials

No data were used to support this study.

Authors' contributions

The author has read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).