Continuity of Pseudo-Differential Operator $h_{\mu,a}$ Involving Hankel Translation and Hankel Convolution on Some Gevrey Spaces

TL;DR

This paper investigates the continuity properties of a specific pseudo-differential operator linked to Bessel operators, involving Hankel translation and convolution, within certain Gevrey spaces, highlighting its boundedness and regularity-preserving features.

Contribution

It establishes the continuity of the pseudo-differential operator $h_{,a}$ on Gevrey spaces involving Hankel translation and convolution, under growth conditions on the symbol's derivatives.

Findings

The operator $h_{,a}$ is continuous on Gevrey spaces.

Hankel translation and convolution preserve Gevrey regularity.

The study provides conditions for the boundedness of the operator.

Abstract

The Pseudo-Differential Operator (p.d.o.) $h_{\mu,a}$ associated with the Bessel Operator involving the symbol $a(x,y)$ whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. $h_{\mu,a}$ on Hankel translation $\tau$ and Hankel convolution of Gevrey functions is continuous linear map into another Gevrey spaces.