Morrey Spaces Associated to the Sections and Singular Integrals
Received:August 07, 2016  Revised:February 27, 2017
Key Word: Morrey space   Campanato space   Monge-Amp\ere singular integral
Fund ProjectL:Supported by Young Foundation of Education Department of Hubei Province (Grant No.Q20162504).
 Author Name Affiliation Songbai WANG College of Mathematics and Statistics, Hubei Normal University, Hubei 435002, P. R. China Jinqiu WEN College of Mathematics and Statistics, Hubei Normal University, Hubei 435002, P. R. China
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In this paper, we define the Morrey spaces $\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$ and the Campanato spaces $\mathcal E^{p,q}_\mathcal F(\mathbb R^n)$ associated with a family $\mathcal F$ of sections and a doubling measure $\mu$, where $\mathcal F$ is closely related to the Monge-Amp\ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to $\mathcal F,$ Monge-Amp\`ere singular integral operators and fractional integrals on $\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$. We also prove that the Morrey spaces $\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$ and the Campanato spaces $\mathcal E^{p,q}_\mathcal F(\mathbb R^n)$ are equivalent with $1\leq q\leq p<\infty$.