Variable Anisotropic Singular Integral Operators

Abstract

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 et al. (Constr Approx 31:149–194, 2010). 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 Bownik (Mem Am Math Soc 164(781):1–122, 2003). 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 et al. (J Fourier Anal Appl 17:1066–1107, 2011). In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range $$0<p\le 1$$ .