Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron Kerman; Luboš Pick

Studia Mathematica (2008)

  • Volume: 186, Issue: 2, page 127-160
  • ISSN: 0039-3223

Abstract

top
We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space W m , ϱ ( Ω ) be compactly imbedded into the rearrangement-invariant space L σ ( Ω ) , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from L ϱ ( 0 , | Ω | ) into L σ ( 0 , | Ω | ) . The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.

How to cite

top

Ron Kerman, and Luboš Pick. "Compactness of Sobolev imbeddings involving rearrangement-invariant norms." Studia Mathematica 186.2 (2008): 127-160. <http://eudml.org/doc/286659>.

@article{RonKerman2008,
abstract = {We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^\{m,ϱ\}(Ω)$ be compactly imbedded into the rearrangement-invariant space $L_\{σ\}(Ω)$, where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_\{ϱ\}(0,|Ω|)$ into $L_\{σ\}(0,|Ω|)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.},
author = {Ron Kerman, Luboš Pick},
journal = {Studia Mathematica},
keywords = {Sobolev imbedding; rearrangement invariant norm; interpolation; Orlicz spaces},
language = {eng},
number = {2},
pages = {127-160},
title = {Compactness of Sobolev imbeddings involving rearrangement-invariant norms},
url = {http://eudml.org/doc/286659},
volume = {186},
year = {2008},
}

TY - JOUR
AU - Ron Kerman
AU - Luboš Pick
TI - Compactness of Sobolev imbeddings involving rearrangement-invariant norms
JO - Studia Mathematica
PY - 2008
VL - 186
IS - 2
SP - 127
EP - 160
AB - We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m,ϱ}(Ω)$ be compactly imbedded into the rearrangement-invariant space $L_{σ}(Ω)$, where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{ϱ}(0,|Ω|)$ into $L_{σ}(0,|Ω|)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.
LA - eng
KW - Sobolev imbedding; rearrangement invariant norm; interpolation; Orlicz spaces
UR - http://eudml.org/doc/286659
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.