Variable Hardy Spaces

Abstract

We develop the theory of variable exponent Hardy spaces Hp(·). We give equivalent definitions in terms of maximal operators that are analogous to the classical theory. We also show that Hp(·) functions have an atomic decomposition including a “finite” decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky [28] than the classical atomic decomposition. As an application of the atomic decomposition, we show that singular integral operators are bounded on Hp(·) with minimal regularity assumptions on the exponent p(·).