Sara Chari, Joshua Frisch, Daniel J. Kelleher, Luke G. Rogers
We prove existence of a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gasket fractals and deduce Gaussian heat kernel bounds in the geodesic metric. Our proof differs from that given by Kigami for the usual Sierpinski gasket in that we show the geodesics are de Rham curves, for which there is an extensive regularity theory.
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We study the analogue of a convex interpolant of two sets on Sierpinski gaskets and an associated notion of measure transport. The structure of a natural family of interpolating measures is described and an interpolation inequality is established. A key tool is a good description of geodesics on these gaskets, some results on which have previously appeared in the literature.
Mean value properties of harmonic functions on Sierpinski gasket type fractals
本文建立了Sierpinski垫片上Laplacian域上的调和函数和一些一般函数的经典均值性质的一个类比。此外，我们将结果推广到了其它的p.c.f.具有二面角-3对称性的分形。
We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L2 domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.
The Power Law For Buffon's Needle Landing Near the Sierpinski Gasket
在本文中，我们得到了一个功率估计，从上面的概率，布冯的针将降落在距离3^{-n}的谢尔宾斯基垫片的豪斯多夫维数1。在比较的情况下1/4角康托尔集考虑纳扎罗夫，佩雷斯，和第二作者：我们仍然需要的技术arXiv：0801.2942分裂的方向好和坏的，但情况下谢尔宾斯基垫片是相当普遍的，缺乏对称性，导致需要更仔细的估计零点的傅里叶变换的康托措施。
Exact Spectral Asymptotics on the Sierpinski Gasket
分形上的分析比流形上的分析更复杂的一个原因是谱计数函数$N(t)$的渐近行为具有由非常数乘法周期函数调制的幂律。然而，我们表明，Sierpinski垫片是可能的，写一个确切的公式，没有剩余项，有效的几乎每一个$t$。这是一个比在流形上有效的结果更强的结果。
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $μ$ we show there is a full $μ$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gra…
We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.
Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket
我们使用频谱抽取提供公式计算的拉普拉斯本征函数的Sierpinski垫片上的谐波梯度。这些公式是根据定义为无穷乘积的特殊函数给出的。
Buffon needles landing near Sierpinski gasket
本文对Nazarov，Peres，和Volberg的“四角康托尔集的Buffon针概率的幂律“（arXiv：0801.2942）方法进行了修正，得到了豪斯多夫维数为1的$n$ -th部分构造Sierpinski垫片的Buffon针概率的估计.
We present a direct proof of the non-degeneracy of the harmonic structures on the level-$n$ Sierpinski gaskets for any $n\geq 2$, which was conjectured by Hino in [H1,H2] and confirmed to be true by Tsougkas [T] very recently using Tutte's spring theorem.
Singular integrals on Sierpinski gaskets
在$d$维、$d <1$、Sierpinski垫片$E_d$上构造了一类与齐次Calderón-Zygmund标准核相关的奇异积分算子.这些算子在$L^2(μ_d)$中有界，并且它们的主值几乎处处发散$μ_d$，其中$μ_d$是$E_d$上的自然（d维）测度。
Orthogonal Polynomials on the Sierpinski Gasket
在包含Sierpinski gasket（{ \bf $SG$ }）的一类分形上构造Laplacian引起了分形分析的深入研究。例如，一个完整的理论多项式和幂级数的$SG$已经制定了我们和他的合著者之一。我们建立在这个身体的工作，以构建某些类似物的经典正交多项式（OP）的$SG$。特别是，我们调查的关键属性，这些OP的$SG$，包括一个三项递推公式和渐近的系数出现在这个递归。此外，我们开发的数值工具，使我们能够图形的一些这些OP。最后，我们使用这些数值工具来调查这些多项式的零和节点集的结构。
We study the analogue of polynomials (solutions to $Δ^{n+1} u =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain "derivative" is 1 at one of the boundary points, while all other "derivatives" vanish, and we compute the values of the monomials at the bo…
Boundary Value Problems for a Family of Domains in the Sierpinski Gasket
对于Sierpinski垫片中的一类区域，我们研究了有限能量的调和函数，用它们的边界值来刻画它们，并研究了它们在边界上的法向导数.我们刻画了那些域，其中有一个扩展运营商的功能有限的能量。我们给出了这些域的格林函数的一个明确的结构。
We study higher order tangents and higher order Laplacians on p.c.f. self-similar sets with fully symmetric structures, such as $D3$ or $D4$ symmetric fractals. Firstly, let $x$ be a vertex point in the graphs that approximate the fractal, we prove that for any $f$ defined near $x$, the higher oder weak tangent of $f$ at $x$, if exists, is the uniform limit of local multiharmonic functions that agree with $f$ in some sense near $x$. Secondly, we prove that the higher order Laplacian on a fractal can be expressible as a renormalized uniform limit of higher order graph Laplacians on the graphs t…
We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop. The proof involves a new technique of constructing an exceptional homeomorphism from $\mathbb R^2$ into some non-planar surface $S$, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem arXiv:math/0107171. We also prove that all homeomorphic copies of the Sierpiński gasket are non-removable for continuous Sobolev functions of the class $W^{1,p}$ for $1\leq p\leq 2$, thus complementing and sharpening the results of the …
Extensions and their Minimizations on the Sierpinski Gasket
研究Sierpinski Gasket（$SG$）上的扩张问题。在第一部分中，我们考虑在有限点集上极小化具有指定值的泛函$\mathcal{E}_λ(f) = \mathcal{E}(f,f) + λ\int f^2 d μ$，其中$\mathcal{E}$表示能量（在欧氏空间中类似于$\int |\nabla f|^2$），$μ$表示$SG$上的标准自相似测度。我们显式地构造了一些常数$c_i$的极小元$f(x) = \sum_{i} c_i G_λ(x_i, x)$，其中$G_λ$是$λ\geq 0$的预解式。我们通过计算极小元$f$的显式二次型$\mathcal{E}(f)$来极小化$SG$中的能量。我们考虑这个二次型的任意集合的性质，然后分析一些特定的集合。我们考虑的一个这样的集合是$SG$的图形近似的底行。我们描述的二次型和离散形式的Haar函数，对应于连续的结果建立在以前的文件。在第二部分中，我们研究了一个类似的问题，这一次最小化$\int_{SG} |Δf(x)|^2 d μ(x)$一般措施。在这两种情况下，通过使用标准的方法，我们证明了存在性和唯一性的最小化问题。然后，我们研究的唯一极小的性质。
Buffon needle lands in $ε$-neighborhood of a 1-Dimensional Sierpinski Gasket with probability at most $|\logε|^{-c}$
近年来，通过研究“投影多重性”函数$f_θ$的$L^p$范数，在Besicovitch投影定理的精神下，对于自相似集已经获得了相对尖锐的定量结果，其中$f_θ(x)$是在$θ$方向上正交投影以覆盖$x$的部分分形集的连通分量的数目。在\cite{NPV}中，我们证明了对于$p<1/6$，具有自相似比例因子1/4的$n$次部分四角Cantor集在Favard长度上的衰减至少与$\frac{C}{n^p}$一样快.在\cite{BV}中，对于某些$p>0$的1维Sierpinski垫片证明了相同的估计。为了使\cite{NPV}的方法适用于垫片，需要进行一些观察：我们在这里绘制了这些观察结果的草图。我们还制定了一个结果，所有自相似集的1维。
Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
本文首次研究Sierpinski垫片上的非线性抛物方程，建立了含两个相互奇异测度$L^p$范数的新型Sobolev不等式，推导了解的时空正则性，并得到了垫片上Burgers方程解的最大值原理、存在唯一性及正则性。
Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series
我们研究了Sierpinski垫片上的幂级数展开的模拟，用于基于Kigami Laplacian的分析。多项式的类似物是多调和函数，其先前已经与泰勒近似和样条一起研究过。这里的主要技术结果是对单项式的大小的估计，类似于x^n/n！。我们提出了一个定义，整个解析函数的功能表示的幂级数，其系数满足指数增长性的条件是强于所需的保证一致收敛性。我们提出了一个表征这些功能的指数增长性条件的权力的拉普拉斯函数。这些完整的解析函数具有人们所期望的性质，例如重排和由无限喷流唯一确定。然而，并非所有的指数函数（拉普拉斯算子的本征函数）都是完全解析的，并且许多其他自然的候选者，例如热核，也不属于这一类。然而，我们能够使用谱抽取来研究指数，特别是创建负特征值的指数衰减函数。
$L^p$-Integrability of Radon-Nikodym Densities Between Harmonic Energy Measures on the Sierpinski Gasket
研究谢尔宾斯基垫片上非常数调和函数能量测度间拉东 - 尼科迪姆密度的$L^p$可积性，通过证明密度比幂和一致有界，得出在特定子区间及特定条件下的$L^p$可积性结论。
In this paper, we focus on Strichartz's derivatives, a family of derivatives including the normal derivative, on p.c.f. (post critically finite) fractals, which are defined at vertex points in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives of the neighboring vertices will decay to zero. The optimal rates of approximations are described and several non-trivial examples are provided to ill…
Measurable Envelopes, Hausdorff Measures and Sierpiński Sets
证明了$\RR^n$的所有子集关于$d$维Hausdorff测度$(0<d<n)$的可测包络的存在性与$ZFC$无关.我们还研究了关于$d$维Hausdorff测度的Sierpienski集可测性的存在性的一致性。
A trace theorem for Dirichlet forms on fractals
考虑了自相似集上的自相似Dirichlet型到自相似子集的迹定理。特别是，我们刻画的域的Dirichlet形式的Sierpinski垫片和Sierpinski地毯的边界，其中边界意味着三角形和矩形的限制垫片和地毯的痕迹。作为一个应用程序，我们构造扩散过程的分形集合称为分形场，表现为适当的分形扩散内的每个分形分量的字段。
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