Removable singularities for Lipschitz caloric functions in time varying domains

刻画Carnot群中各类函数空间可移除集的容量

Capacities Characterizing Removable Sets for Various Function Spaces in Carnot Groups

本文研究了Carnot群中各类函数空间的可移除集，通过不同容量和算子刻画其可移除性及Lipschitz函数特性。

抛物型Dini-β条件与曲面测度及热测度的绝对连续性

The parabolic Dini-$β$ condition and absolute continuity of surface and caloric measure

该研究证明抛物型Lipschitz函数图像的曲面测度与热测度的绝对连续性等价于抛物型Dini-β条件成立，且这类图像可被正则Lipschitz图像覆盖，为抛物型可求长性提供了定性判定依据。

The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions

This work provides a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{dist(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset \mathbb R^{n+1}$. As a corollary, one o…

Lipschitz调和函数的里斯变换、可求直性和可移性

The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions

我们证明了，给定一个具有有限$n$ -Hausdorff测度$H^n$的集合$E\subset \mathbb R^{n+1}$，如果$n$ -维里斯变换$$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$在$L^2(H^n|E)$中有界，则$E$是$n$ -可求积的.从这个结果我们推出了一个紧集$E\subset\mathbb R^{n+1}$与$H^n(E)<\infty$是Lipschitz调和函数可去的当且仅当它是纯粹的$n$ -不可求长的，从而证明了Vitushkin猜想在高维的类比.

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。

具有$L^2$-有界Riesz变换的测度与Painlevé问题

The measures with $L^2$-bounded Riesz transform and the Painlevé problem

本文刻画了$\mathbb R^{n+1}$中具有$n$次多项式增长的测度使得$n$维Riesz变换属于$L^2(\mu)$的几何条件，并由此得到Lipschitz调和函数可去集的度量几何刻画及其在双Lipschitz映射下的不变性。