Variable Hardy spaces D. Cruz-UribeDengyin Wang 42B2542B3042B35.Hardy spacesvariable Lebesgue spacesgrand maximal operatoratomic decompositionsingular integral operators 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 $p(\cdot)$. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5232 10.1512/iumj.2014.63.5232 en Indiana Univ. Math. J. 63 (2014) 447 - 493 state-of-the-art mathematics http://iumj.org/access/