$L^p$-bounds for pseudo-differential operators on compact Lie groups

Abstract

Given a compact Lie group $G$, in this paper we establish $L^p$-bounds for pseudo-differential operators in $L^p(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\widehat{G}$, where $\widehat{G}$ is the unitary dual of $G$. We obtain two different types of $L^p$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{ρ,δ}^m(G)$ classes which are a suitable extension of the well known $(ρ,δ)$ ones on the Euclidean space. The results herein extend classical $L^p$ bounds established by C. Fefferman on $\mathbb R^n$. While Fefferman's results have immediate consequences on general manifolds for $ρ>\max\{δ,1-δ\}$, our results do not require the condition $ρ>1-δ$. Moreover, one of our results also does not require $ρ>δ$. Examples are given for the case of SU(2)$\cong\mathbb S^3$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^m$ and $\mathscr{S}_{\frac12,0}^m$ naturally appear, and where conditions $ρ>δ$ and $ρ>1-δ$ fail, respectively.