Local approximations and intrinsic characterizations of spaces of smooth functions on regular subsets of $R^n$

Characterization of traces of smooth functions on Ahlfors regular sets

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.

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto-Piepenbrinck type theorem, which is based on a ground state transfor…

Regularity versus smoothness of measures

The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by considering the $L^p$ norms of its density. We establish sharp relationships between these two notions. Roughly speaking, we show that smooth measures must be regular, but that regular measures need not be smooth.

On temporal regularity of stochastic convolutions in $2$-smooth Banach spaces

We show that paths of solutions to parabolic stochastic differential equations have the same regularity in time as the Wiener process (as of the current state of art). The temporal regularity is considered in the Besov-Orlicz space $B^{1/2}_{Φ_2,\infty}(0,T;X)$ where $Φ_2(x)=\exp(x^2)-1$ and $X$ is a $2$-smooth Banach space.

Heat kernels and regularity for rough metrics on smooth manifolds

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

分布非负内蕴曲率的弱正则曲面的凸性

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

本文证明了在 $C^{1,α}$（$α>2/3$）正则性及分布 Gaussian 曲率非零非负条件下，二维完备 Riemannian 流形等距嵌入 ${\mathbb R}^3$ 的像为凸曲面，其分析依赖于对极弱 Monge-Ampère 方程解的关键观察。