New L^p-boundedness properties for the Kontorovich–Lebedev and Mehler–Fock transforms

ABSTRACT

In this paper, the authors present a systematic study of several new $L^p$-boundedness properties for the Kontorovich–Lebedev transform and the Mehler–Fock transform on the spaces $L_{\gamma ,p}\left({\mathbb{R}}_{+}\right)$ $(1\leqq p<\mathrm{\infty };\gamma \in \mathbb{R})$ and $L^{\mathrm{\infty }}\left({\mathbb{R}}_{+}\right)$. Relevant connections with various earlier related results are also pointed out.

KEYWORDS:

• Kontorovich–Lebedev transform
• Mehler–Fock transform
• L^p-boundedness properties
• modified Bessel (or the Macdonald) function
• associated Legendre function
• Hölder's inequality

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 44A15
• Secondary: 46E30

Acknowledgements

The authors wish to thank the referee for several helpful and invaluable suggestions for the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.