## Hardy spaces with variable exponents on RD-spaces and applications

### Volume 520 / 2016

Dissertationes Mathematicae 520 (2016), 1-74
MSC: Primary 42B30; Secondary 42B25, 42B20, 42B35, 47B06, 30L99.
DOI: 10.4064/dm744-9-2015
Published online: 28 October 2016

#### Abstract

In this article, the authors introduce Hardy spaces with variable exponents, $H^{\ast,p(\cdot)}(\mathcal X)$, on RD-spaces with infinite measures via the grand maximal function. Then the authors characterize these spaces by means of the non-tangential maximal function or the dyadic maximal function. Characterizations in terms of atoms or Littlewood–Paley functions are also established. As applications, the authors prove an Olsen inequality for fractional integral operators and the boundedness of singular integral operators and quasi-Banach valued sublinear operators on these spaces. Finally, a duality theory of these spaces is developed.