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This paper is concerned with developing a theory of traces for functions thatare integrable but need not possess any differentiability within their domain.Moreover, the domain can have an irregular boundary with cusp-like features andcodimension not necessarily equal to one, or even an integer. Given$\Omega\subseteq\mathbb{R}^n$ and $\Gamma\subseteq\partial\Omega$, we introducea function space $\mathscr{N}^{s(\cdot),p}(\Omega)\subseteqL^p_{\text{loc}}(\Omega)$ for which a well-defined trace operator can beidentified. Membership in $\mathscr{N}^{s(\cdot),p}(\Omega)$ constrains theoscillations in the function values as $\Gamma$ is approached, but does notimply any regularity away from $\Gamma$. Under connectivity assumptions between$\Omega$ and $\Gamma$, we produce a linear trace operator from$\mathscr{N}^{s(\cdot),p}(\Omega)$ 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 upperAhlfors-regular, then the trace is a continuous operator into aSobolev-Slobodeckij space. If $\Gamma=\partial\Omega$ and is further assumed tobe lower Ahlfors-regular, then the trace exhibits the standard Lebesgue pointproperty. 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 onlytangentially accessible.

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