Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities
Ron Kerman, Luboš Pick (2011)
Studia Mathematica
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We study imbeddings of the Sobolev space : = u: Ω → ℝ with < ∞ 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, and , that are optimal with respect to the inclusions . General formulas for and are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.