$L^p$-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 $G\times \hat{G}$, where $\hat{G}$ 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 ${\mathcal{S}}_{\rho ,\delta }^m(G)$ classes which are a suitable extension of the well known $(\rho ,\delta )$ 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 $\rho >\max \{\delta ,1-\delta \}$, our results do not require the condition $\rho >1-\delta$. Moreover, one of our results also does not require $\rho >\delta$. Examples are given for the case of SU(2)$\cong {\mathbb{S}}^3$ and vector fields/sub-Laplacian operators when operators in the classes ${\mathcal{S}}_{0,0}^m$ and ${\mathcal{S}}_{\frac{1}{2},0}^m$ naturally appear, and where conditions $\rho >\delta$ and $\rho >1-\delta$ fail, respectively.

29 pages; a section on motivation and applications is added; to appear in J. Inst. Math. Jussieu

References in corpus (2)

• Wave equation for sums of squares on compact Lie groups
• Net spaces on lattices, Hardy-Littlewood type inequalities, and their converses

Cited by in corpus (1)

• Hölder-Besov boundedness for periodic pseudo-differential operators