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 $u\in L^{ 1,\infty}(SU(2)).$
Categories: math.AP