Morrey Spaces Associated to the Sections and Singular Integrals

Abstract:

      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 $1\leq q\leq p<\infty$.