Variable Besov and Triebel--Lizorkin spaces.
We study different definitions of the first order variable exponent Sobolev space with zero boundary values in an open subset of .
The trace space of consists of those functions on ℝⁿ that can be extended to functions of (as in the fixed-exponent case). Under the assumption that p is globally log-Hölder continuous, we show that the trace space depends only on the values of p on the boundary. In our main result we show how to define an intrinsic norm for the trace space in terms of a sharp-type operator.
In a recent article the authors showed that it is possible to define a Sobolev capacity in variable exponent Sobolev space. However, this set function was shown to be a Choquet capacity only under certain assumptions on the variable exponent. In this article we relax these assumptions.