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Fuglede’s theorem in variable exponent Sobolev space. (English) Zbl 1070.46023

Summary: 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 \(p:\Omega\to[m,M]\) for \(1<m\leq M<\infty\), and smooth functions are dense in Sobolev space.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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