# Partial spectral multipliers and partial Riesz transforms for degenerate operators

### A. F. M. ter Elst

University of Auckland, New Zealand### El Maati Ouhabaz

Université Bordeaux 1, Talence, France

## Abstract

We consider degenerate differential operators of the type $A=−∑_{k,j=1}∂_{k}(a_{kj}∂_{j})$ on $L_{2}(R_{d})$ with real symmetric bounded measurable coefficients. Given a function $χ∈C_{b}(R_{d})$ (respectively, a bounded Lipschitz domain $Ω$), suppose that $(a_{kj})≥μ>0$ a.e. on $suppχ$ (respectively, a.e. on $Ω$). We prove a spectral multiplier type result: if $F:[0,∞)→C$ is such that $sup_{t>0}∥φ(.)F(t.)∥_{C_{s}}<∞$ for some nontrivial function $φ∈C_{c}(0,∞)$ and some $s>d/2$ then $M_{χ}F(I+A)M_{χ}$ is weak type (1,1) (respectively, $P_{Ω}F(I+A)P_{Ω}$ is weak type (1,1)). We also prove boundedness on $L_{p}$ for all $p∈(1,2]$ of the partial Riesz transforms $M_{χ}∇(I+A)_{−1/2}M_{χ}$. The proofs are based on a criterion for a singular integral operator to be weak type (1,1).

## Cite this article

A. F. M. ter Elst, El Maati Ouhabaz, Partial spectral multipliers and partial Riesz transforms for degenerate operators. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 691–713

DOI 10.4171/RMI/735