A sharp iteration principle for higher-order Sobolev embeddings

• Autorzy: Andrea Cianchi , Luboš Pick , Lenka Slavíková
• Języki publikacji: EN

Abstrakty

EN

We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving arbitrary rearrangement-invariant norms on open sets in ℝⁿ, possibly endowed with a measure density and satisfying an isoperimetric inequality of fairly general type, to considerably simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a direct application of the reduction principle we determine the optimal target space in the relevant Sobolev embeddings both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, the results apply to any-order Sobolev embedding on regular (John) domains, on Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and on families of product probability spaces, of which the Gauss space and the exponential measure space are classical instances.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2014
• Tom: 101
• Numer: 1
• Strony: 37-58

Daty

wydano

2014

Twórcy

autor

Andrea Cianchi

• Dipartimento di Matematica e Informatica "U.Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

autor

Luboš Pick

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

autor

Lenka Slavíková

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Identyfikatory

DOI

10.4064/bc101-0-3