Partial spectral multipliers and partial Riesz transforms for degenerate operators

TL;DR

This paper establishes spectral multiplier results and boundedness of partial Riesz transforms for degenerate differential operators with measurable coefficients, extending harmonic analysis tools to degenerate elliptic operators.

Contribution

It introduces new weak type (1,1) spectral multiplier theorems and boundedness results for partial Riesz transforms for degenerate operators with minimal regularity.

Findings

Spectral multipliers are weak type (1,1) under smoothness conditions.

Partial Riesz transforms are bounded on L^p for p in (1,2].

Results extend harmonic analysis to degenerate elliptic 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 $\chi \in C_b^\infty(\mathbb{R}^d)$ (respectively, $\Omega$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge \mu > 0$ a.e.\ on $ \supp \chi$ (resp., a.e.\ on $\Omega$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| \varphi(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $\varphi \in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_\chi F(I+A) M_\chi$ is weak type $(1,1)$ (resp.\ $P_\Omega F(I+A) P_\Omega$ 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 $(1,1)$.