Canonical potential and L^p-Sobolev space involving linear canonical Fourier transform

Abstract

The main objective of this paper is to enrich the theoretical system of the linear canonical Fourier transform (LCFT) by introducing the canonical potential $J_{m,A}$ and corresponding $L^p$-Sobolev space. Moreover, the Schwartz-type space $S_A$ is introduced. Further, pseudo-differential operator (PDO) is defined and obtained its another integral representation. The $L^p$-boundedness result for the pseudo-differential operator associated with the LCFT is discussed. Some applications of Sobolev-type spaces ${\mathcal{H}}_A^{m,p}$ and $H_A^{m,p}$ are given.

Keywords:

• Linear canonical Fourier transform
• Schwartz-type space
• canonical potential
• Sobolev space
• convolution

AMS CLASSIFICATIONS:

• 43A32
• 46F05
• 35S05
• 46E35
• 44A35

Acknowledgments

Authors are very thankful to the reviewers for his/her valuable and constructive comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Department of Science andTechnology, Ministry of Science and Technology, india , under grant  no. DST/INSPIRE Fellowship/2017/IF170292.