SG-Hankel Pseudo-Differential Operators on Weighted Gelfand-Shilov Type Spaces and a Numerical Example

TL;DR

This paper introduces a new class of SG pseudo-differential operators linked with the Hankel transform on weighted Gelfand-Shilov spaces, analyzing their properties, compactness, and solvability, with a numerical example demonstrating decay behavior.

Contribution

It defines the SG--Hankel symbol class and studies the associated pseudo-differential operators' properties, including continuity, compactness, and solvability, on weighted Gelfand-Shilov spaces.

Findings

Operators are continuous on weighted Gelfand-Shilov spaces.

Under decay conditions, operators exhibit compactness between weighted spaces.

Numerical example confirms theoretical decay predictions.

Abstract

We introduce a new class of SG pseudo-differential operators associated with the Hankel transform on a family of weighted Gelfand--Shilov type spaces of radial functions. First, we recall basic properties of the Hankel transform of order $\nu>-1/2$ and define a convenient Gelfand--Shilov type space $W_{\alpha,\beta}$ which is invariant under the Hankel transform and stable under differentiation and multiplication by powers of the radial variable. Then we define the SG--Hankel symbol class $S^{m_1,m_2}_H$ and the corresponding pseudo-differential operator \[ (T_\sigma f)(x)=\int_0^\infty \sigma(x,\lambda)J_\nu(x\lambda)\widehat f_H(\lambda)\,\lambda\,d\lambda. \] We prove that $T_\sigma$ is continuous on $W_{\alpha,\beta}$, and under additional decay assumptions on the symbol, we obtain compactness results between different weighted spaces. Minimal and maximal realisations of $T_\sigma$ in $L^2((0,\infty),x\, dx)$ are studied in detail, and a weak solvability result for the SG--Hankel pseudo-differential equation $T_\sigma f=g$ is derived. Finally, we present a numerical example for a simple SG symbol and a Gaussian input, illustrating the spatial decay predicted by the theory.