Variable Anisotropic Singular Integral Operators

We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $\Theta$ of $\mathbb{R}^n$ introduced by Dahmen, Dekel, and Petrushev \cite{ddp}. This is an extension of the classical isotropic singular integral operators on $\mathbb{R}^n$ of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author \cite{b}. We establish the boundedness of variable anisotropic singular integral operators $T$ on the Hardy spaces with pointwise variable anisotropy $H^p(\Theta)$, which were developed by Dekel, Petrushev, and Weissblat \cite{dpw}. In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range $0<p\leq 1$.