Asian Journal of Mathematics

Volume 20 (2016)

Number 1

Characterization of Campanato spaces associated with parabolic sections

Pages: 183 – 198



Ming-Yi Lee (Department of Mathematics, National Central University, Taiwan)

Chin-Cheng Lin (Department of Mathematics, National Central University, Taiwan)

Xinfeng Wu (Department of Mathematics, China University of Mining & Technology, Beijing, China)


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.


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

2010 Mathematics Subject Classification

42B30, 42B35

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