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

• Autorzy: Ron Kerman , Luboš Pick
• Języki publikacji: EN

Abstrakty

EN

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 𝓚-method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 206
• Numer: 2
• Strony: 97-119

Daty

wydano

2011

Twórcy

autor

Ron Kerman

• Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1

autor

Luboš Pick

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

Identyfikatory

DOI

10.4064/sm206-2-1