迭代Big-Pieces运算符和较大集合
Iterating the Big--Pieces operator and larger sets
摘要：
我们证明了如果一个维数为$k$的Ahlfors-David正则集$E$有大片Lipschitz图（通常用$BP(BP(LG))$表示），则$E\subset \tilde{E}$，其中$\tilde{E}$是维数为$k$的Ahlfors-David正则集，并且有大片Lipschitz图（通常用$BP(LG)$表示）.我们的结果是定量的，事实上，被证明在一个度量空间的任何家庭的Ahlfors-David定期集$\mathcal{F}$取代$LG$。一个简单的推论是BP算子在2次迭代后的稳定性。这是以前只知道在欧几里德设置的情况下$\mathcal{F}= LG$与实质上更复杂的证明。
英文摘要：
We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting for the case $\mathcal{F}= LG$ with substantially more complicated proofs.
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