Pseudo-differential operator associated with the linear canonical Dunkl transform

Abstract

In this paper, we introduce a pseudo-differential operator associated with the linear canonical Dunkl transform. For a particular class of symbols, we prove that this operator defines a continuous linear mapping from the Schwartz space into itself. We further establish an amplitude integral and kernel representation of the operator and investigate the smoothness and decay properties of the associated kernel. Subsequently, we establish an $L^1$-norm inequality for the pseudo-differential operator on the linear canonical Dunkl Sobolev spaces. Moreover, we extend the pseudo-differential operator to tempered distributions, proving its continuity. We then investigate the $L^2$-boundedness of the operator for the symbol class $S_0^m$. Finally, as an application, we employ the pseudo-differential operator to study non-homogeneous and nonlinear parabolic partial differential equations.