Partial spectral multipliers and partial Riesz transforms for degenerate operators

Abstract

We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $χ\in C_b^\infty(\mathbb{R}^d)$ (respectively, $Ω$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge μ> 0$ a.e.\ on $ \supp χ$ (resp., a.e.\ on $Ω$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| φ(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $φ\in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_χF(I+A) M_χ$ is weak type $(1,1)$ (resp.\ $P_ΩF(I+A) P_Ω$ is weak type $(1,1)$). We also prove boundedness on $L^p$ for all $p \in (1,2]$ of the partial Riesz transforms $M_χ\nabla (I + A)^{-1/2}M_ χ$. The proofs are based on a criterion for a singular integral operator to be weak type $(1,1)$.