SG-Hankel Pseudo-Differential Operators on Weighted Gelfand-Shilov Type Spaces and a Numerical Example
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2025
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| _version_ | 1866917229907410944 |
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| author | Pasawan, Durgesh |
| author_facet | Pasawan, Durgesh |
| contents | 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 $ν>-1/2$ and define a convenient Gelfand--Shilov type space $W_{α,β}$ 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 \[ (T_σf)(x)=\int_0^\infty σ(x,λ)J_ν(xλ)\widehat f_H(λ)\,λ\,dλ. \] We prove that $T_σ$ is continuous on $W_{α,β}$, and under additional decay assumptions on the symbol, we obtain compactness results between different weighted spaces. Minimal and maximal realisations of $T_σ$ in $L^2((0,\infty),x\, dx)$ are studied in detail, and a weak solvability result for the SG--Hankel pseudo-differential equation $T_σf=g$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SG-Hankel Pseudo-Differential Operators on Weighted Gelfand-Shilov Type Spaces and a Numerical Example Pasawan, Durgesh Functional Analysis 2020: 35S05, 46F12, 47G30, 65R10 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 $ν>-1/2$ and define a convenient Gelfand--Shilov type space $W_{α,β}$ 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 \[ (T_σf)(x)=\int_0^\infty σ(x,λ)J_ν(xλ)\widehat f_H(λ)\,λ\,dλ. \] We prove that $T_σ$ is continuous on $W_{α,β}$, and under additional decay assumptions on the symbol, we obtain compactness results between different weighted spaces. Minimal and maximal realisations of $T_σ$ in $L^2((0,\infty),x\, dx)$ are studied in detail, and a weak solvability result for the SG--Hankel pseudo-differential equation $T_σf=g$ 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. |
| title | SG-Hankel Pseudo-Differential Operators on Weighted Gelfand-Shilov Type Spaces and a Numerical Example |
| topic | Functional Analysis 2020: 35S05, 46F12, 47G30, 65R10 |
| url | https://arxiv.org/abs/2601.00800 |