- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:47
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
29
2013
2
Partial spectral multipliers and partial Riesz transforms for degenerate operators
A. F. M.
ter Elst
University of Auckland, AUCKLAND, NEW ZEALAND
E.
Ouhabaz
Université Bordeaux 1, TALENCE CEDEX, FRANCE
Spectral multipliers, Riesz transforms, singular integral operators, degenerate operators, Gaussian bounds
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).
Fourier analysis
Integral equations
General
691
713
10.4171/RMI/735
http://www.ems-ph.org/doi/10.4171/RMI/735