Pseudo-differential operators associated with the fractional Hankel-Bessel transform

TL;DR

This paper develops a new class of pseudo-differential operators linked to a fractional Hankel-Bessel transform, extending classical Hankel analysis with fractional and global perspectives, including symbol classes and boundedness results.

Contribution

It introduces a fractional Hankel-Bessel pseudo-differential calculus with new symbol classes, kernel estimates, and boundedness properties, expanding the analytical framework for such transforms.

Findings

Established boundedness on weighted L^p-spaces.

Derived kernel estimates and integral representations.

Defined global Shubin-type symbol classes for the fractional transform.

Abstract

We introduce and study a new class of pseudo-differential operators associated with a fractional Hankel--Bessel transform. Motivated by the classical Hankel transform and the pseudo-differential operators associated with Bessel operators studied by Pathak and Pandey \cite{PathakPandey1995}, we define a fractional variant by inserting a fractional Fourier-type phase into the Hankel kernel. We then introduce global Shubin-type symbol classes adapted to this transform, derive kernel estimates and integral representations, and establish boundedness results on weighted L^{p}-spaces and on fractional Hankel--Sobolev spaces. This provides a new framework parallel to the classical Hankel pseudo-differential calculus, but in a fractional and global setting.