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

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 -