IUMJ

Title: Variable Hardy spaces

Authors: David Cruz-uribe and Daniel Wang

Issue: Volume 63 (2014), Issue 2, 447-493

Abstract:

We develop the theory of variable exponent Hardy spaces $H^{p(\cdot)}$. We give equivalent definitions in terms of maximal operators that are analogous to the classical theory. We also show that $H^{p(\cdot)}$ functions have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Str\"omberg and Torchinsky [J.\:O. Str\"omberg and A. Torchinsky, \textit{Weighted Hardy Spaces}, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989] than the classical atomic decomposition. As an application of the atomic decomposition, we show that singular integral operators are bounded on $H^{p(\cdot)}$ with minimal regularity assumptions on the exponent $p(\cdot)$.