论文标题
非整合方形函数的二次稀疏支配和加权估计值
Quadratic sparse domination and Weighted Estimates for non-integral Square Functions
论文作者
论文摘要
我们证明了一般非综合方函数$ s $的二次稀疏支配结果。也就是说，我们证明了形式的估计 \ begin {equation*} \ int_ {m}（s f）^{2} g \，\ mathrm {d}μ\ le c \ sum_ {p \ in \ mathcal {s}} \ left（\ frac {1} {\ lvert 5p \ rvert} \ int_ {5 p} \ lvert f \ rvert^{p_ {0}}} \，\ mathrm {d}μ\ right） \ left（\ frac {1} {\ lvert 5p \ rvert} \ int_ {5 p} \ lvert g \ rvert^{q__ {0}^*}^*} \，\ mathrm {d}μ\ right） \ lvert p \ rvert， \ end {equation*}其中$ q_ {0}^{*} $是$ q_ {0}/2 $，$ m $的Hölder共轭，是基础倍增空间，$ \ Mathcal {s} $是$ m $ $ m $的立方体的稀疏集合。我们的结果将涵盖与发散形式椭圆算子以及与Laplace-Beltrami操作员相关的正方形功能。这种稀疏的统治使我们能够在加权空间$ l^{p}（w）$中得出最佳的规范估计。
We prove a quadratic sparse domination result for general non-integral square functions $S$. That is, we prove an estimate of the form \begin{equation*} \int_{M} (S f)^{2} g \, \mathrm{d}μ\le c \sum_{P \in \mathcal{S}} \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} \, \mathrm{d}μ\right)^{2/p_{0}} \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert g\rvert^{q_{0}^*}\,\mathrm{d}μ\right)^{1/q_{0}^*} \lvert P\rvert, \end{equation*} where $q_{0}^{*}$ is the Hölder conjugate of $q_{0}/2$, $M$ is the underlying doubling space and $\mathcal{S}$ is a sparse collection of cubes on $M$. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $L^{p}(w)$.
