2020/07/02 by Mikil Foss, Foss, Mikil
Computer Science · Mathematics · #35A23 #46E35 #47G10 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.00863
openalex publication_date 2020/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with developing a theory of traces for functions that\nare integrable but need not possess any differentiability within their domain.\nMoreover, the domain can have an irregular boundary with cusp-like features and\ncodimension not necessarily equal to one, or even an integer. Given\n\Ω\⊆\ℝn and \Γ\⊆\∂\Ω, we introduce\na function space mathscrNs(\⋅),p(\Ω)\⊆\nLp\loc(\Ω) for which a well-defined trace operator can be\nidentified. Membership in mathscrNs(\⋅),p(\Ω) constrains the\noscillations in the function values as \Γ is approached, but does not\nimply any regularity away from \Γ. Under connectivity assumptions between\n\Ω and \Γ, we produce a linear trace operator from\n mathscrNs(\⋅),p(\Ω) to the space of measurable functions on\n\Γ. The connectivity assumptions are satisfied, for example, by all\n1-sided nontangentially accessible domains. If \Γ is upper\nAhlfors-regular, then the trace is a continuous operator into a\nSobolev-Slobodeckij space. If \Γ=\∂\Ω and is further assumed to\nbe lower Ahlfors-regular, then the trace exhibits the standard Lebesgue point\nproperty. To demonstrate the generality of the results, we construct\n\Ω\⊆\ℝ2 with a t>1-dimensional Ahlfors-regular\n\Γ\⊆\∂\Ω satisfying the main domain hypotheses, yet\n\Γ is nowhere rectifiable and for every neighborhood of every point in\n\Γ, there exists a boundary point within that neighborhood that is only\ntangentially accessible.\n