Pseudo-Differential Operators and $\mathfrak{T}$- Wigner Function on Locally Compact Communicative Hausdorff Groups (Journal of Nonlinear Modeling and Analysis, 2024)
In this article, we consider a harmonic analysis of locally compactgroups and introduce a generalization of the classical cross-Wigner distributiondefined on $G × \hat{G}$ by $$W_{\mathfrak{F}}(ψ,φ)(g,ξ)=\int_{G}\overline{ξ(h)}ψ(τ_1(g,h))\overline{φ(τ_2(g,h))}dμ(h).$$We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show thatassume $Λ$ and $Γ$ are closed cocompact subgroups of $G$ and $\hat{G}$, respectively,then, for a given window $φ∈ L^2(G),$ either both systems $\{m_{γ}τ_{λ}φ\}_{λ\inΛ,γ\inΓ}$ and $\{m_{κ}τ_vφ\}_{κ\inΛ^{⊥},v\inΓ^{⊥}}$ are Gabor systems in $L^2(G),$ simultaneously, withthe same upper bound, or neither $\{m_{γ}τ_{λ}φ\}_{λ\inΛ,γ\inΓ}$ nor $\{m_{κ}τ_vφ\}_{κ\inΛ^{⊥},v\inΓ^{⊥}}$ comprises a Gabor system. Finally, pseudo-differential operators on locallycompact groups are studied, we establish that assuming a pseudo-differentialoperator $A_a$ corresponds to the symbol $a\in W^{∞,1}_{τ,1oι^{-1}}(G×\hat{G})$ then $A_a$ isbounded operator $W^{p,q}_{τ}(G)→ W^{p,q}_{τ}(G)$.
Keywords: Fourier transform; Wigner function; compact group; pseudo-differential operator; symbol; Weyl-Heisenberg frame.