  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.
