Boundary trace inequalities and rearrangements

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived. This research was partially supported by the Italian research project “Geometric properties of solutions to variational problems” of GNAMPA (INdAM) 2006, by the research project MSM 0021620839 of the Czech Ministry of Education, by grants 201/03/0935, 201/05/2033 and 201/07/0388 of the Grant Agency of the Czech Republic and by the Nečas Center for Mathematical Modeling project no. LC06052 financed by the Czech Ministry of Education.