Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman; Luboš Pick

Studia Mathematica (2011)

  • Volume: 206, Issue: 2, page 97-119
  • ISSN: 0039-3223

Abstract

top
We study imbeddings of the Sobolev space W m , ϱ ( Ω ) : = u: Ω → ℝ with ϱ ( α u / x α ) < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, τ ϱ and σ ϱ , that are optimal with respect to the inclusions W m , ϱ ( Ω ) W m , τ ϱ ( Ω ) L σ ϱ ( Ω ) . General formulas for τ ϱ and σ ϱ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

How to cite

top

Ron Kerman, and Luboš Pick. "Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities." Studia Mathematica 206.2 (2011): 97-119. <http://eudml.org/doc/285787>.

@article{RonKerman2011,
abstract = {We study imbeddings of the Sobolev space $W^\{m,ϱ\}(Ω)$: = u: Ω → ℝ with $ϱ(∂^\{α\}u/∂x^\{α\})$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_\{ϱ\}$ and $σ_\{ϱ\}$, that are optimal with respect to the inclusions $W^\{m,ϱ\}(Ω) ⊂ W^\{m,τ_\{ϱ\}\}(Ω) ⊂ L_\{σ_\{ϱ\}\}(Ω)$. General formulas for $τ_\{ϱ\}$ and $σ_\{ϱ\}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.},
author = {Ron Kerman, Luboš Pick},
journal = {Studia Mathematica},
keywords = {Sobolev imbedding; rearrangement-invariant norm; optimal range norm; optimal hull norm; Köthe dual; K-method of interpolation; Brudnyi-Krugljak duality theory; Lorentz gamma norm; Orlicz norm; integral operator; supremum operator},
language = {eng},
number = {2},
pages = {97-119},
title = {Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities},
url = {http://eudml.org/doc/285787},
volume = {206},
year = {2011},
}

TY - JOUR
AU - Ron Kerman
AU - Luboš Pick
TI - Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities
JO - Studia Mathematica
PY - 2011
VL - 206
IS - 2
SP - 97
EP - 119
AB - We study imbeddings of the Sobolev space $W^{m,ϱ}(Ω)$: = u: Ω → ℝ with $ϱ(∂^{α}u/∂x^{α})$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_{ϱ}$ and $σ_{ϱ}$, that are optimal with respect to the inclusions $W^{m,ϱ}(Ω) ⊂ W^{m,τ_{ϱ}}(Ω) ⊂ L_{σ_{ϱ}}(Ω)$. General formulas for $τ_{ϱ}$ and $σ_{ϱ}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.
LA - eng
KW - Sobolev imbedding; rearrangement-invariant norm; optimal range norm; optimal hull norm; Köthe dual; K-method of interpolation; Brudnyi-Krugljak duality theory; Lorentz gamma norm; Orlicz norm; integral operator; supremum operator
UR - http://eudml.org/doc/285787
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.