Morrey Spaces Associated to the Sections and Singular Integrals

DOI：10.3770/j.issn:2095-2651.2017.04.007

 作者 单位 王松柏 湖北师范大学数学与统计学院, 湖北 黄石 435002 温金秋 湖北师范大学数学与统计学院, 湖北 黄石 435002

设$\mathcal F$是与Monge-Amp\ere紧密相关的一族截面, $\mu$是一个双倍测度, 我们定义了与之相关的Morre空间$\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$和Campanato空间$\mathcal E^{p,q}_\mathcal F(\mathbb R^n).$ 并且, 我们得到了与截面族$\mathcal F$相关的Hardy-Littlewood极大算子, 奇异积分和分数次积分在Morre空间$\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$的有界性. 我们也证明了, 当$1\leq q\leq p<\infty$时, Morre空间$\mathcal M^{p,q}_\mathcal F(\mathbb R^n)$和Campanato空间$\mathcal E^{p,q}_\mathcal F(\mathbb R^n)$是等价的.

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 $1\leq q\leq p<\infty$.