Characterization of Campanato spaces associated with parabolic sections

We study the Campanato spaces Λ^k _q,P associated with a family P of parabolic sections which are closely related to the parabolic Monge-Ampère equation. We characterize these spaces in terms of Lipschitz spaces Li_P^α P. We also introduce the corresponding Hardy spaces H^p _P and demonstrate the equivalence between the Littlewood-Paley g-functions and atomic decompositions for elements in H^p _P. Moreover, we show that Campanato spaces are the duals of Hardy spaces.