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Optimal Sobolev trace embeddings

Authors: Andrea Cianchi and Luboš Pick
Journal: Trans. Amer. Math. Soc. 368 (2016), 8349-8382
MSC (2010): Primary 46E35, 46E30
Published electronically: January 19, 2016
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Abstract: Optimal target spaces are exhibited in arbitrary-order Sobolev type embeddings for traces of $ n$-dimensional functions on lower dimensional subspaces. Sobolev spaces built upon any rearrangement-invariant norm are allowed. A key step in our approach consists of showing that any trace embedding can be reduced to a one-dimensional inequality for a Hardy type operator depending only on $ n$ and on the dimension of the relevant subspace. This can be regarded as an analogue for trace embeddings of a well-known symmetrization principle for first-order Sobolev embeddings for compactly supported functions. The stability of the optimal target space under iterations of Sobolev trace embeddings is also established and is part of the proof of our reduction principle. As a consequence, we derive new trace embeddings, with improved (optimal) target spaces, for classical Sobolev, Lorentz-Sobolev and Orlicz-Sobolev spaces.

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Andrea Cianchi
Affiliation: Dipartimento di Matematica e Informatica \lq\lq U. Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

Luboš Pick
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Keywords: Sobolev spaces, trace embeddings, optimal target, rearrangement-invariant spaces, Orlicz spaces, Lorentz spaces, supremum operators
Received by editor(s): April 24, 2014
Received by editor(s) in revised form: September 27, 2014
Published electronically: January 19, 2016
Additional Notes: This research was partly supported by the research project Prin 2008 “Geometric aspects of partial differential equations and related topics” of MIUR (Italian Ministry of University), by GNAMPA of the Italian INdAM (National Institute of High Mathematics), and by the grants 201/08/0383 and P201/13/14743S of the Grant Agency of the Czech Republic.
Article copyright: © Copyright 2016 American Mathematical Society