Singular integrals on Ahlfors-David regular subsets of the Heisenberg group

On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1

We prove that if $μ$ is a d-dimensional Ahlfors-David regular measure in $\R^{d+1}$, then the boundedness of the $d$-dimensional Riesz transform in $L^2(μ)$ implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of $μ$.

Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.

Ahlfors正则平坦边界上调和测度的维数下降失效

Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries

该研究通过修改David-Jeznach-Julia的构造，证明对任意s∈(0,1/2)存在对应Ahlfors正则紧集，结合已有结果确定调和测度维数下降的阈值1/2是尖锐的。

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on t…

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

Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we …

Uniform measures and uniform rectifiability

In this paper it is shown that if $μ$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $μ$ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in $R^d$. In particular, it is shown here that they satisfy the "big pieces of Lipschitz graphs" property.

Singular sets for harmonic measure on locally flat domains with locally finite surface measure

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $Ω\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partialΩ)<\infty$ and a subset $E\subset \partial Ω$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimens…

关于抛物空间和海森堡群中的奇异积分与定量可测性

On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group

本文将David和Semmes关于奇异积分与定量可测性的定理扩展到抛物空间和海森堡群。

The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected …

多重分形障碍物的积分方程方法

Integral Equation Methods for Scattering by Multifractal Obstacles

本文研究了多重分形障碍物的声学散射问题，提出了一种基于积分方程的方法，解决了具有空间变化分形维度的更现实的分形散射体的散射场表示问题，并建立了收敛的Galerkin方法。