Integrability of the Fourier–Jacobi transform of functions satisfying Lipschitz and Dini–Lipschitz-type estimates

ABSTRACT

The purpose of this work is to prove analogues of the classical Titchmarsh theorem and Younis' theorem associated with the Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition of a certain order in suitable weighted spaces $L^p\left(\right[0,+\mathrm{\infty }\left)\right)$, $1<p\le 2$. For this purpose, we use a generalized translation operator defined by Flensted-Jensen and Koornwinder.

Keywords:

• Fourier–Jacobi operator
• Fourier–Jacobi transform
• generalized translation operator
• Lipschitz class
• Dini–Lipschitz class
• Titchmarsh's theorem

2010 Mathematics Subject Classifications:

• 33C45
• 41A17

Acknowledgements

The authors are grateful to the referees for the useful comments and suggestions in improving the presentation of the paper.

Disclosure statement

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