Mikil Foss
University of Nebraska-Lincoln, Lincoln, USA

Given a uniformly continuous function on an open domain, there is a unique continuous extension to the boundary. The trace operator provides a function that captures the boundary values of this extension. Gagliardo’s trace theorem extends this concept to the Sobolev spaces. There have since been many generalizations and refinements of Gagliardo’s theorem. Typically, these trace theorems require some differentiability of the function within its domain and some regularity of the domain’s boundary. These assumptions ensure there is a well-defined boundary value function. Moreover, this trace will, itself, possess some differentiability and a certain Lebesgue point property. I will present a trace theorem that provides a well-defined boundary-value function that exists in a fractional Sobolev space and has the Lebesgue point property yet requires no differentiability within the domain and allows very irregular boundaries. The result is motivated by boundary-value problems involving nonlocal operators that are defined for integrable but not necessarily differentiable functions.