论文标题
较弱的下密度条件和均匀的可区分性
The weak lower density condition and uniform rectifiability
论文作者
论文摘要
我们表明，如果$（x，\ infty）$，ahlfors $ d $ d $ regratiard set $ e $ in $ \ mathbb {r}^{n} $如果对$（x，r）\ in E \ times（0，\ infty）$，则在b（x，r）$ y \ y \ y \ y \ y \ y \ y \ y \ y \（x，r）$ 0 < $ \ MATHSCR {h}^{d} _ {\ infty}（e \ cap b（y，t））<（2t）^{d} - \ varepsilon（2r）^d $是每个$ \ varepsilon> 0 $的Carleson设置。 为了证明这一点，我们通过证明$ x $是一个$ c $ floupsring space，$ \ varepsilon，ρ\ in（0,1）$，$ a> 1 $，而$ x_ {n} $是最大$ 2^{ - n} $ x $ x $ x $ x $ x $ x $ x $ x $ $ \ mathscr {b} = \ {b（x，2^{ - n}）：x \ in x_ {n}，n \ in \ Mathbb {n} \} $，然后\ sum \ [\ sum \ sum \ sum \ lest \ lest \ lest \ {r_ {r_ {b}^{b}^{s} \ frac {\ mathscr {h}^{s} _ {ρr_{ρr_{b}}（x \ cap ab）} {（2r_ {b}） \ Mathscr {h}^{s}（x）。 \]这是经典结果的定量版本，对于公制空间$ x $的有限$ s $数二维的hausdorff测度，上$ s $二维密度最多是$ 1 $ $ $ \ mathscr {h}^{s}^{s} $几乎到处都是。
We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0<t<r$ satisfying $\mathscr{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\varepsilon(2r)^d$ is a Carleson set for every $\varepsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\varepsilon,ρ\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathscr{B}=\{B(x,2^{-n}):x\in X_{n},n\in \mathbb{N}\}$, then \[ \sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{ρr_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\varepsilon\right\} \lesssim_{C,A,\varepsilon,ρ,s} \mathscr{H}^{s}(X). \] This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\mathscr{H}^{s}$-almost everywhere.
