Pseudo-Differential Operators and $\mathfrak{T}$- Wigner Function on Locally Compact Communicative Hausdorff Groups

Abstract

In this article, we consider a harmonic analysis of locally compact groups and introduce a generalization of the classical cross-Wigner distribution defined on $G × \hat{G}$ by $$W_{\mathfrak{F}}(\psi,\varphi)(g,\xi)=\int_{G}\overline{\xi(h)}\psi(\tau_1(g,h))\overline{\varphi(\tau_2(g,h))}d\mu(h).$$We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show that assume $\Lambda$ and $\Gamma$ are closed cocompact subgroups of $G$ and $\hat{G}$, respectively, then, for a given window $\phi∈ L^2(G),$ either both systems $\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$ and $\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$ are Gabor systems in $L^2 (G),$ simultaneously, with the same upper bound, or neither $\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$ nor $\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$ comprises a Gabor system. Finally, pseudo-differential operators on locally compact groups are studied, we establish that assuming a pseudo-differential operator $A_a$ corresponds to the symbol $a\in W^{\infty,1}_{\tau,1oι^{-1}}(G\times\hat{G})$ then $A_a$ is bounded operator $W^{p,q}_{\tau}(G)\rightarrow W^{p,q}_{\tau}(G)$.