Lizaveta Ihnatsyeva, Antti V. Vähäkangas
We extend the results of P. Shvartsman on characterizing the traces of Besov and Triebel-Lizorkin spaces on Ahlfors $n$-regular sets to the case of $d$-regular sets, $n-1<d<n$. The characterizations of trace spaces are given in terms of local polynomial approximations.
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In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
We prove that functions in the Hajłasz-Sobolev space $M^{1,s}$ on an $s$-Ahlfors regular metric space are uniformly continuous when $s\le 1$.
On a quasi-additivity formula for the capacity on Ahlfors-regular spaces
该研究在树结构边界与紧Ahlfors正则空间框架下证明了径向型核对应容量的拟可加性公式，还定义了相关函数的单变量调和延拓并证明其边界切向收敛性。
Let $n\ge2$ and $ϕ: [0,\fz) \to [0,\infty)$ be a Young's function satisfying $\sup_{x>0} \int_0^1\frac{ϕ( t x)}{ ϕ(x)}\frac{dt}{t^{n+1} }<\infty. $ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${\dot {\bf B}}^ϕ$ extension domains, which is necessary to guarantee the nontrivially of ${\dot {\bf B}}^ϕ$. On the other hand, assume that $ϕ$ grows sub-exponentially at $\fz$ additionally. If $Ω$ is a Besov-Orlicz ${\dot {\bf B}}^ϕ$ extension domain, then it must be Ahlfors $n$-regular.
Traces of Sobolev spaces to piecewise Ahlfors--David regular sets
本文在支持弱局部 Poincaré 不等式的加倍度量测度空间中，刻画 Sobolev 空间到不同余维数的有限个 Ahlfors–David 正则子集并集上的迹，并给出迹范数的定量估计。
Traces of weighted Besov spaces to Ahlfors-David regular sets: the limiting case
针对加权Besov空间到Ahlfors-David正则集的迹问题，引入弱化正则性条件，给出了迹空间的完整内蕴描述，推广了Haroske和Schmeisser的结果。
Decomposition of functions of bounded variation in Ahlfors-regular PI spaces
本研究针对非紧Ahlfors正则PI空间，将有界变差函数分解为可数个有界变差单调函数之和，且总变差满足可加性，把欧氏空间的相关结果推广到了度量空间情形。
Pointwise estimates for rough operators in a metric measure framework under some Ahlfors regularity conditions
本文在Ahlfors正则条件下，研究了度量测度框架中粗糙算子的点估计，并推导出相关功能不等式。
Hölder regularity of the solutions of Fredholm integral equations on upper Ahlfors regular sets
本文研究了在满足上Ahlfors增长条件的度量测度空间中，Fredholm二类积分方程解的Hölder正则性问题。
Remez Type Inequalities and Morrey-Campanato Spaces on Ahlfors Regular Sets
本文给出了Lebesgue n-测度为零的Ahlfors正则子集上n元实多项式和复多项式的Remez型不等式的几个新结果。作为应用，我们证明了定义在这类集合上的Morrey-Campanato空间的一个扩张定理。
Heat kernel for reflected jump diffusion on Ahlfors regular domains
本文在阿赫弗斯正则域上研究反射跳跃扩散的热核估计，通过构造扩展算子并建立混合稳定型热核估计，探讨了反射过程在一般度量测度空间中的行为特性。
Descriptions of traces of weighted Sobolev spaces to Ahlfors--David regular sets in the case $p=1$
针对 $p=1$ 的加权 Sobolev 空间，给出了其在 Ahlfors-David 正则集上迹空间的完全内蕴描述，并构造了非线性及有界线性延拓算子。
The trace-free Beurling--Ahlfors transform and the Bourgain--Brezis problem for Hodge systems
该研究引入无迹Beurling--Ahlfors变换，利用其符号的反射奇性得到端点抵消估计，证明了任意维度与形式次数下Bourgain--Brezis猜想的希尔伯特情形，还得到多类新的临界估计及端点Hodge分解，并证明多数情形下有界选择算子非线性。
Riesz Transform Characterizations of $H^1$ and {\rm BMO} on Ahlfors Regular Sets with Small Oscillations
该研究利用Riesz变换，对带小振荡的Ahlfors正则集上的Hardy空间$H^1$和BMO空间给出刻画，推广了平坦欧氏空间中的经典结果。
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices $s<1,$ as well as the first order Hajłasz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined intrinsically on $F$. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
Ahlfors regularity of continua that minimize maxitive set functions
本文在抽象最大性条件下证明平面连续统中集函数极小化子具有Ahlfors正则性，并给出长度与半径可比的定量估计，统一涵盖谱泛函、内半径和扭转函数最大值等应用对象。
The logarithmic Dirichlet Laplacian on Ahlfors regular spaces
本研究在Ahlfors正则度量测度空间上内蕴定义了对数Laplace-Beltrami算子，阐明了其热半群、定义域等谱性质，并证明其与非等距同胚作用在非交换几何意义下相容。
Traces of Subharmonic Functions to Fractal Sets
研究了一类下调和函数到$\Co^{n}$的Ahlfors正则子集的迹.特别地，我们建立了迹的一个推广的BMO性质和逆的赫尔德不等式.
Traces of Besov spaces to regular subsets of metric measure spaces: the limiting case
研究度量测度空间中Besov空间在Ahlfors-David余维θ正则子集上的迹，证明了迹算子有界性，并构造了非线性延拓算子，将迹空间识别为L_p空间。
We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of the Euclidean space and the relation between these spaces and traces of classical Sobolev spaces.
Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},μ)$ be a metric measure space with uniformly locally doubling measure $μ$ supporting a weak local $(1,p)$-Poincaré inequality. For each $θ\in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension-$θ$ content regular closed subsets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},μ)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we get an intrinsic description of the trace-space of the Sobolev $W^{1}_{p…
We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.
We present a survey of recent results of the functional analysis allowing to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a d-set boundary on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and t…
Let $(X,d,μ)$ be an Ahlfors metric measure space. We give sufficient conditions on a closed set $F\subseteq X$ and on a real number $β$ in such a way that $d(x,F)^β$ becomes a Muckenhoupt weight. We give also some illustrations to regularity of solutions of partial differential equations and regarding some classical fractals.
Embeddings of variable Sobolev, Besov, and Triebel-Lizorkin spaces on metric measure spaces
本文研究了在度量测度空间上可变指数Hajłasz型光滑空间的嵌入理论，提出新的可变指数Sobolev、Triebel-Lizorkin和Besov空间，并建立了到可变指数Lebesgue和Hölder空间的Sobolev、Morrey和Moser-Trudinger型嵌入。
Large-scale behaviour of Sobolev functions in Ahlfors regular metric measure spaces
本文研究 Ahlfors 正则度量测度空间中 p-Sobolev 函数在无穷远处的极限行为，引入 p-瘦集与 Q-厚集刻画极限存在性，并给出函数增长速率界。
We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness…
Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures
针对带奇异测度的矩阵Birman-Schwinger算子，对紧Lipschitz曲面上自然测度建立其奇异数和特征值的Weyl型渐近公式，对一般Ahlfors正则测度证明先前上谱估计为阶最优。
Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
该论文通过结合无迹 Beurling--Ahlfors 变换与非线性对偶方法，在 Besov 和 Triebel--Lizorkin 空间中扩展了 Hodge 系统的 Bourgain--Brezis 选择映射的适用范围，显著拓宽了已知解范围。
A Fermionic Grunsky operator
本文针对边界正则的共形映射引入了费米子 Grunsky 算子，证明其拉回的 Smirnov 空间为算子图像，并刻画了其满足 Hilbert-Schmidt 条件时与 Weil-Petersson 拟圆盘的等价关系。
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