Kernel estimates and weak (1,1)-boundedness of pseudo-differential operators on compact Lie groups

Abstract

Given a compact Lie group $G$ and its unitary dual $\widehat{G}$, we establish the weak (1,1) continuity for pseudo-differential operators in the global H\"ormander classes of order $-n(1-\rho)/2$ on $G\times \widehat{G}$. Our approach consists of proving suitable estimates for the kernel of such operators. Furthermore, we use these kernel estimates to give an alternative proof for the $H^1(G)$-$L^1(G)$-continuity of these classes now allowing the full range $0\leq\delta\leq\rho\leq1, \;\rho\neq0,\;\delta\neq1$. The conditions for the operators are formulated using the H\"ormander classes $S^m_{\rho,\delta}(G):=S^m_{\rho,\delta}(G\times \widehat{G})$ of symbols in the non-commutative phase space $G\times \widehat{G}$, which are extensions of the well-known $(\rho,\delta)$-classes in the Euclidean space. Our results are formulated in the complete range $0\leq \delta\leq \rho\leq 1,$ $\rho\neq0,\;$$\delta\neq 1$. As an application of this boundedness result we provide end-point a-priori $L^1$-estimates for the sub-Laplacian $\mathcal{L}_{sub}=X^2+Y^2,$ and for the heat type operator $T=Z-X^2-Y^2$ on $SU(2)\cong \mathbb{S}^3$ that cannot be obtained by application of the standard pseudo-differential calculus due to H\"ormander. More precisely, we prove that if one considers the subelliptic problem, \begin{equation}\label{IVP:abstract} \begin{cases}Tu=f ,& \text{ } \\u,f\in \mathscr{D}'(SU(2)):=(C^\infty(SU(2)))', & \text{ } \end{cases} \end{equation} then, for $f\in W^{1,-\frac{1}{4}}(SU(2)),$ one has that $u\in L^{1,\infty}(SU(2)).$