Weighted Hardy inequalities beyond Lipschitz domains

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent \lambda=n-1. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.Comment: 12 pages, to appear in Proc. Amer. Math. So

Measures with predetermined regularity and inhomogeneous self-similar sets

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding self-similar set $E$ and the condensation set $C$. If the Assouad dimension of $C$ is strictly smaller than the Assouad dimension of $E$, then the upper regularity dimension of any measure supported on $E_C$ is strictly larger than the Assouad dimension of $E_C$. Surprisingly, the corresponding statement for the lower regularity dimension fails

Uniqueness of diffusion on domains with rough boundaries

Let $\Omega$ be a domain in $\mathbf R^d$ and $h(\varphi)=\sum^d_{k,l=1}(\partial_k\varphi, c_{kl}\partial_l\varphi)$ a quadratic form on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the $c_{kl}$ are real symmetric $L_\infty(\Omega)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in \Omega$. Further assume there are $a, \delta>0$ such that $a^{-1}d_\Gamma^{\delta}\,I\le C\le a\,d_\Gamma^{\delta}\,I$ for $d_\Gamma\le 1$ where $d_\Gamma$ is the Euclidean distance to the boundary $\Gamma$ of $\Omega$. We assume that $\Gamma$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $\Gamma$, is larger or equal to $d-1$ we also assume a mild uniformity property for $\Omega$ in the neighbourhood of one $z\in\Gamma$. Then we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if $\delta\ge 1+(s-(d-1))$. The result applies to forms on Lipschitz domains or on a wide class of domains with $\Gamma$ a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in $\mathbf R^2$ or the complement of a uniformly disconnected set in $\mathbf R^d$.Comment: 25 pages, 2 figure