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

TL;DR

This paper develops a trace theory for integrable functions in irregular domains with complex boundaries, extending classical results to functions with no differentiability and boundaries with fractal-like features.

Contribution

It introduces a new function space and establishes trace operators for functions with minimal regularity on irregular, possibly fractal, boundaries in alculus.

Findings

Defined a trace operator for alculus functions on irregular boundaries.

Proved continuity of the trace into Sobolev-Slobodeckij spaces under Ahlfors-regularity.

Constructed examples with fractal boundaries satisfying main hypotheses.

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 $\mathscr{N}^{s(\cdot),p}(\Omega)\subseteq L^p_{\text{loc}}(\Omega)$ for which a well-defined trace operator can be identified. Membership in $\mathscr{N}^{s(\cdot),p}(\Omega)$ 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 $\mathscr{N}^{s(\cdot),p}(\Omega)$ 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 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.