嵌套分形上 BV 测度的振荡

Good labeling property of simple nested fractals

We show various criteria to verify if a given nested fractal has a good labeling property, inter alia we present a characterization of GLP for fractals with an odd number of essential fixed points. We show a convenient reduction of area to be investigated in verification of GLP and give examples that further reduction is impossible. We prove that if a number of essential fixed points is a power of two, then a fractal must have GLP and that there are no values other than primes or powers of two guaranteeing GLP. For all other numbers of essential fixed points we are able to construct examples h…

嵌套分形上的 $L^p$-Poincaré 不等式

$L^p$-Poincaré inequalities on nested fractals

本文利用热核方法发展分形上 Korevaar-Schoen-Sobolev 空间的局部 $L^p$ 理论，证明了嵌套分形上度量球的尺度不变 $L^p$-Poincaré 不等式，并应用于 Sobolev 不等式、极大函数及 Hajłasz-Sobolev 空间研究。

具有垂直缩放函数的分形插值曲面的盒维数

Box dimension of fractal interpolation surfaces with vertical scaling function

针对广义仿射分形插值曲面，本文提出借助振荡和、振荡向量及垂直缩放矩阵建立层级关系，并在特定约束下确定其盒维数。

紧集与相对分形鼓的分形管公式：振荡、复维数与分形性

Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality

本文为任意维欧几里得空间中的相对分形鼓建立逐点与分布分形管公式，用分形 ζ 函数的复维数留数表达邻域体积，并提出基于非实复维数或自然边界的新分形性定义。

Box dimension of generalized affine fractal interpolation functions

Let $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $\dim_B Γf$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By introducing vertical scaling matrices, we estimate the upper bound and the lower bound of oscillations of $f$. As a result, we obtain explicit formula of $\dim_B Γf$ under certain constraint conditions.

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isope…