Partial spectral multipliers and partial Riesz transforms for degenerate operators

Abstract

We consider degenerate differential operators of the type $A=-{\sum }_{k,j=1}^d{\partial }_k(a_{kj}{\partial }_j)$ on $L^2({\mathbb{R}}^d)$ with real symmetric bounded measurable coefficients. Given a function $\chi \in C_b^{\infty }({\mathbb{R}}^d)$ (respectively, a bounded Lipschitz domain $\Omega$), suppose that $(a_{kj})\ge \mu >0$ a.e. on $\mathrm{supp}\chi$ (respectively, a.e. on $\Omega$). We prove a spectral multiplier type result: if $F:[0,\infty )\to \mathbb{C}$ is such that ${\sup }_{t>0}\Vert \phi (.)F(t.){\Vert }_{C^s}<\infty$ for some nontrivial function $\phi \in C_c^{\infty }(0,\infty )$ and some $s>d/2$ then $M_{\chi }F(I+A)M_{\chi }$ is weak type (1,1) (respectively, $P_{\Omega }F(I+A)P_{\Omega }$ is weak type (1,1)). We also prove boundedness on $L^p$ for all $p\in (1,2]$ of the partial Riesz transforms $M_{\chi }\nabla (I+A{)}^{-1/2}{M}_{\chi }$. The proofs are based on a criterion for a singular integral operator to be weak type (1,1).