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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 1

### Characterization of Campanato spaces associated with parabolic sections

Pages: 183 – 198

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n1.a8

#### Authors

#### Abstract

We study the Campanato spaces $\Lambda^{\kappa}_{q, \mathcal{P}}$ associated with a family $\mathcal{P}$ of parabolic sections which are closely related to the parabolic Monge–Ampère equation. We characterize these spaces in terms of Lipschitz spaces $\mathrm{Lip}^{\alpha}_{\mathcal{P}}$. We also introduce the corresponding Hardy spaces $H^{p}_{\mathcal{P}}$ and demonstrate the equivalence between the Littlewood-Paley $g$-functions and atomic decompositions for elements in $H^{p}_{\mathcal{P}}$. Moreover, we show that Campanato spaces are the duals of Hardy spaces.

#### Keywords

Campanato spaces, Hardy spaces, Lipschitz spaces, Monge–Ampère equations, parabolic sections

#### 2010 Mathematics Subject Classification

42B30, 42B35

Published 28 January 2016