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

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 $\nu >-1\mathrm{/}2$ 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 $S_H^{m_1,m_2}$ and the corresponding pseudo-differential operator $$(T_{\sigma }f)(x)={\int }_0^{\mathrm{\infty }}\sigma (x,\lambda )J_{\nu }(x\lambda ){\widehat{f}}_H(\lambda )\lambda d\lambda \mathrm{.}$$ 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 $L^2((0,\mathrm{\infty }),xdx)$ 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.