Smooth operators for the action of \(SO(3)\) on \(L^2(S^2)\) (Q1378113)

The authors consider the class of the linear bounded operators \(A\) on \(L^2(\mathbb{S}^2)\). Basing on the eigenfunctions of the Laplace operator \(\Delta\) on \(\mathbb{S}^2\), they represent \(A\) by a formula of pseudo-differential type, where the signature of \(A\) plays the role of the symbol. Given \(g\in\text{SO}(3)\), let us consider the unitary operators on \(L^2(\mathbb{S}^2)\) defined by \[ T_gu(x)= u(g^{-1}x). \] The main theorem of the paper roughly says that the mapping \[ g\in\text{SO}(3)\to {_gA}= T_g AT^{-1}_g \] is smooth in norm topology, if and only if the signature is a smooth function. The authors then consider the class of pseudo-differential operators with smooth signature, and give precise estimates of their norms in \(L^2(\mathbb{S}^2)\).