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

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 $S_H^{m_1,m_2}$ and the corresponding pseudo-differential operator \[ (T_σf)(x)=\int_0^\infty σ(x,λ)J_ν(xλ)\widehat f_H(λ)\,λ\,dλ. \] 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 $L^2((0,\infty ),xdx)$ 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.

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