Revista Matemática Iberoamericana


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Volume 29, Issue 2, 2013, pp. 691–713
DOI: 10.4171/RMI/735

Published online: 2013-04-22

Partial spectral multipliers and partial Riesz transforms for degenerate operators

A. F. M. ter Elst[1] and El Maati Ouhabaz[2]

(1) University of Auckland, New Zealand
(2) Université Bordeaux 1, Talence, France

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 $ \operatorname{supp} \chi$ (respectively, 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 nontrivial 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) (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).

Keywords: Spectral multipliers, Riesz transforms, singular integral operators, degenerate operators, Gaussian bounds

ter Elst A. F. M., Ouhabaz El Maati: Partial spectral multipliers and partial Riesz transforms for degenerate operators. Rev. Mat. Iberoam. 29 (2013), 691-713. doi: 10.4171/RMI/735