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

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 $Ω\subseteq\mathbb{R}^n$ and $Γ\subseteq\partialΩ$, we introduce a function space $\mathscr{N}^{s(\cdot),p}(Ω)\subseteq L^p_{\text{loc}}(Ω)$ for which a well-defined trace operator can be identified. Membership in $\mathscr{N}^{s(\cdot),p}(Ω)$ constrains the oscillations in the function values as $Γ$ is approached, but does not imply any regularity away from $Γ$. Under connectivity assumptions between $Ω$ and $Γ$, we produce a linear trace operator from $\mathscr{N}^{s(\cdot),p}(Ω)$ to the space of measurable functions on $Γ$. The connectivity assumptions are satisfied, for example, by all $1$-sided nontangentially accessible domains. If $Γ$ is upper Ahlfors-regular, then the trace is a continuous operator into a Sobolev-Slobodeckij space. If $Γ=\partialΩ$ 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 $Ω\subseteq\mathbb{R}^2$ with a $t>1$-dimensional Ahlfors-regular $Γ\subseteq\partialΩ$ satisfying the main domain hypotheses, yet $Γ$ is nowhere rectifiable and for every neighborhood of every point in $Γ$, there exists a boundary point within that neighborhood that is only tangentially accessible.

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