Traces on General Sets in ${\mathbb{R}}^n$ for Functions with no Differentiability Requirements

Abstract

This paper is concerned with developing a theory of traces for functions that are integrable but need not possess any differentiability within their domain. Moreover, the domain can have an irregular boundary with cusp-like features and codimension not necessarily equal to one, or even an integer. Given $\Omega \subseteq {\mathbb{R}}^n$ and $\Gamma \subseteq \partial \Omega$, we introduce a function space ${\mathcal{N}}^{s(\cdot ),p}(\Omega )\subseteq L_{\text{loc}}^p(\Omega )$ for which a well-defined trace operator can be identified. Membership in ${\mathcal{N}}^{s(\cdot ),p}(\Omega )$ constrains the oscillations in the function values as $\Gamma$ is approached, but does not imply any regularity away from $\Gamma$. Under connectivity assumptions between $\Omega$ and $\Gamma$, we produce a linear trace operator from ${\mathcal{N}}^{s(\cdot ),p}(\Omega )$ to the space of measurable functions on $\Gamma$. The connectivity assumptions are satisfied, for example, by all $1$-sided nontangentially accessible domains. If $\Gamma$ is upper Ahlfors-regular, then the trace is a continuous operator into a Sobolev-Slobodeckij space. If $\Gamma =\partial \Omega$ and is further assumed to be lower Ahlfors-regular, then the trace exhibits the standard Lebesgue point property. To demonstrate the generality of the results, we construct $\Omega \subseteq {\mathbb{R}}^2$ with a $t>1$-dimensional Ahlfors-regular $\Gamma \subseteq \partial \Omega$ satisfying the main domain hypotheses, yet $\Gamma$ is nowhere rectifiable and for every neighborhood of every point in $\Gamma$, there exists a boundary point within that neighborhood that is only tangentially accessible.