Positivity (2016) 20:435–466
Integrable cross sections in mixed-norm spaces
and Sobolev embeddings
· Javier Soria
Received: 14 February 2015 / Accepted: 7 September 2015 / Published online: 18 September 2015
© Springer Basel 2015
Abstract We characterize optimal Sobolev embeddings in terms of integrable cross
sections and mixed-norm spaces, involving general rearrangement-invariant estimates.
We also ﬁnd the optimal domains and ranges for these embeddings.
Keywords Rearrangement-invariant spaces · Mixed-norm spaces · Embeddings ·
Mathematics Subject Classiﬁcation 46E35 · 46E30
In a recent paper, Algervik and Kolyada  have obtained an extension of the Sobolev
endpoint embedding in terms of integrable cross sections and certain Lorentz norms.
More precisely, they proved that
where the space on the right-hand side is a mixed-norm space (see Deﬁnition 3.2).
N. Clavero and J. Soria have been partially supported by the Grants MTM2013-40985-P
(Spanish Government) and 2014SGR289 (Catalan Autonomous Government).
Department of Applied Mathematics and Analysis, University of Barcelona, Gran Via 585,
08007 Barcelona, Spain