热方程函数与抛物一致可求长集的 Carleson 测度估计

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper establishes equivalence between uniform rectifiability of the boundary of a domain and the property that the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). The results are novel in a variety of ways, in particular (1) this is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; the "direct" result established here is new even in the half space; (2) the results are …

有界调和函数基于Carleson测度估计的一致可求正性和$\varepsilon$ -可逼近性

Uniform rectifiability from Carleson measure estimates and $\varepsilon$-approximability of bounded harmonic functions

设$Ω\subset\mathbb R^{n+1}$，$n\geq1$是具有Ahlfors-David正则边界的螺旋域.本文证明了$\partialΩ$是一致$n$ -可求长的，如果$Ω$上的每一个有界调和函数都是$\varepsilon$ -可逼近的，或者如果$Ω$上的每一个有界调和函数都满足一个适当的平方函数Carleson测度估计.特别地，这适用于当$Ω=\mathbb R^{n+1}\setminus E$和$E$是Ahlfors-David正则的情况。我们的结果解决了霍夫曼，马特尔和梅博罗达在最近的工作中提出的一个猜想，他们证明了相反的陈述。在这里，我们还获得了两个额外的标准一致可求直。一个是在所谓的“$S<N$“的估计，另一个在一个合适的电晕分解涉及谐波措施。

Uniform rectifiability implies Varopoulos extensions

We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson…

Green function estimates on complements of low-dimensional uniformly rectifiable sets

It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the "flagship" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators $L_{β,γ} =- {\rm div} D^{d+1+γ-n} \nabla$ associated to a domain $Ω\subset \mathbb R^n$ with a uniformly rectifiable boundary $Γ$ of dimensi…

Green functions and smooth distances

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function $G$ behaves like a distance function to the boundary, in the sense that $\Big|\frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\Big|^2D(X) dX$ is the density of a Carleson measure, where $D$ is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's $α$-number of uniformly recti…

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem

We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular bo…

通过ACF单调性公式的一致可求正性、椭圆测度、平方函数和$\varepsilon$ -可逼近性

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula

设$Ω\subset\mathbb{R}^{n+1}$，$n\geq2$是具有Ahlfors-David正则边界且满足螺旋条件的开集.我们考虑一个一致椭圆算子$L$的发散形式与矩阵$A$与真正的，只是有界的，可能是非对称的系数，这也是局部Lipschitz和满足适当的Carleson型估计。本文证明了：如果$L^*$是与$A$的转置矩阵相关联的散度型算子，则$\partialΩ$是一致$n$ -可求长的当且仅当$Lu=0$的每一有界解和$Ω$中的$L^*v=0$的每一有界解是$\varepsilon$ -可逼近的当且仅当$Lu=0$的每一有界解和$Ω$中的$L^*v=0$的每一有界解满足适当的平方-函数Carleson测度估计此外，我们得到了两个额外的一致可求直的准则。一个是在所谓的$S<N$估计方面，另一个在一个合适的电晕分解涉及$L$ -谐波和$L^*$ -谐波措施。证明了若$L$ -调和测度和$L^*$ -调和测度满足弱$A_\infty$ -型条件，则$\partial Ω$是$n$ -一致可求长的.在这个过程中，我们得到了一个版本的Alt-Caffarelli-Friedman单调性公式的一个相当广泛的一类椭圆算子，这是独立的利益，并在我们的论点中发挥了重要作用。

高维情形下的Carleson $\u03b5^2$猜想

Carleson's $\varepsilon^2$ conjecture in higher dimensions

本文证明高维版Carleson $\u03b5^2$猜想，给出$\u03b5_n(x,r)$定义并证明对应积分有限的点集的Borel子集为$n$-可求长，还在容量密度条件下刻画了边界切点的积分判据，改进并推广了原猜想。

热方程$L^p$ Dirichlet问题的可解性等价于抛物型Lipschitz图情形下的抛物型一致可修正性

Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

本文证明抛物型Lipschitz图域上热量测度的$A_\infty$性质等价于边界抛物型一致可修正性，从而结合Lewis-Murray工作解决了热方程$L^p$ Dirichlet问题可解性的长期开放问题。

单边弦弧域上椭圆算子的扰动。第一部分：对称算子的小扰动和大扰动

Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

设$Ω\subset\mathbb{R}^{n+1}$，$n\ge 2$是一个单侧弦弧域，即满足内部Corkscrew和Harnack链条件（它们分别是开性和路连通性的尺度不变/定量形式）的域，其边界$\partialΩ$是$n$维Ahlfors正则的.考虑$L_0$和$L$两个实对称散度型椭圆算子，并设$ω_{L_0}$，$ω_L$为相应的椭圆测度.证明了如果$ω_{L_0}\in A_\infty(σ)$，其中$σ=H^n|_{\partialΩ}$，和$L$是$L_0$的扰动（在这个意义上，$L_0$和$L$之间的差异满足一定的Carleson测度条件），则$ω_L\in A_\infty(σ)$ .此外，如果$L$是$L_0$的一个足够小的扰动，那么我们可以保留逆Hölder类，也就是说，如果对于某些$1<p<\infty$，我们有$ω_{L_0}\in RH_p(σ)$，那么就有$ω_{L}\in RH_p(σ)$。等价地，如果$L^{p'}(σ)$中的Dirichlet问题对于$L_0$是可解的，那么对于$L$也是如此。这些结果可以看作是Dahlberg，February man-Kenig-Pipher和Milakis-Pipher-Toro在更良性的环境中得到的扰动定理的推广。作为我们的方法的结果，我们可以证明，对于拉普拉斯算子的任何扰动（或者更一般地说，任何具有满…

图区域上抛物型 Dirichlet 问题 $L^p$-可解性的一个变系数自由边界问题

A Variable Coefficient Free Boundary Problem for $L^p$-solvability of Parabolic Dirichlet Problems in Graph Domains

本文研究了图区域上抛物型 Dirichlet 问题的变系数自由边界问题，证明了在变系数抛物算子满足特定 Carleson 条件下，$L^p$-可解性蕴含图函数在 BMO 中具有半阶时间导数，并克服了平移不变性缺失的挑战。

Elliptic measures and Square function estimates on 1-sided chord-arc domains

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundar…