Hardy Spaces on ℝ n with Pointwise Variable Anisotropy

Abstract

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝn that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝn and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.