Resumen de Fuglede's theorem in variable exponent Sobolev space

Petteri Harjulehto, Peter Hästö, Olli Martio

• Consider an open set $\Omega\subset\mathbb{R}^n$ and a function in the variable exponent Sobolev space $W^{1, p(\cdot)}(\Omega)$. We show that there exists a family of curves $\Gamma$ with zero $p(\cdot)$-modulus such that the quasi-continuous representative of $u$ is absolutely continuous on every rectifiable path not in $\Gamma$. To prove this result we need the following assumptions: the exponent satisfies $p: \Omega\rightarrow [m,M]$ for $1 < m \leq M < \infty $ and smooth functions are dense in Sobolev space.