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Variable exponent trace spaces

Lars DieningPeter Hästö — 2007

Studia Mathematica

The trace space of W 1 , p ( · ) ( × [ 0 , ) ) consists of those functions on ℝⁿ that can be extended to functions of W 1 , p ( · ) ( × [ 0 , ) ) (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.

A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces.

Petteri HarjulehtoPeter Hästö — 2004

Revista Matemática Complutense

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.

Uniform convexity and associate spaces

Petteri HarjulehtoPeter Hästö — 2018

Czechoslovak Mathematical Journal

We prove that the associate space of a generalized Orlicz space L φ ( · ) is given by the conjugate modular φ * even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ -function is equivalent to a doubling Φ -function. As a consequence, we conclude that L φ ( · ) is uniformly convex if φ and φ * are weakly doubling.

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