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.
We study the Poincaré inequality in Sobolev spaces with variable exponent. Under a rather mild and sharp condition on the exponent p we show that the inequality holds. This condition is satisfied e.g. if the exponent p is continuous in the closure of a convex domain. We also give an essentially sharp condition for the exponent p as to when there exists an imbedding from the Sobolev space to the space of bounded functions.
We prove that the associate space of a generalized Orlicz space is given by the conjugate modular even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling -function is equivalent to a doubling -function. As a consequence, we conclude that is uniformly convex if and are weakly doubling.
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.
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