Variable Hardy Spaces

TL;DR

This paper develops the theory of variable exponent Hardy spaces, providing equivalent definitions, atomic decompositions, and establishing boundedness of singular integral operators under minimal regularity conditions.

Contribution

It introduces a new atomic decomposition for variable Hardy spaces similar to weighted Hardy space decompositions, and proves boundedness of singular integrals with minimal assumptions.

Findings

Atomic decomposition includes a finite form similar to weighted Hardy spaces.

Singular integral operators are bounded on these spaces under minimal regularity.

Equivalent definitions via maximal operators are established.

Abstract

We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on variable Hardy spaces with minimal regularity assumptions on the exponent function.