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- Title
ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS.
- Authors
Milman, Mario; Pustylnik, Evgeniy
- Abstract
Let Ω be an open domain in ℝn, let k∈ℕ, $p\le \frac{n}{k}$. Using a natural extension of the L(p, q) spaces and a new form of the Pólya–Szegö symmetrization principle, we extend the sharp version of the Sobolev embedding theorem: $W_0^{k, p} (\Omega)\subset L (\frac{np}{n -kp}, p) to the limiting value $p =\frac{n}{k}$. This result extends a recent result in [3] for the case k=1. More generally, if Y is a r.i. space satisfying some mild conditions, it is shown that $W_0^{k, Y} (\Omega)\subset Y_n (\infty, k) =\{f: t^{-k/n}(f^{\ast\ast} (t)-f^\ast (t))\in Y\}$. Moreover Yn(∞,k) is not larger (and in many cases essentially smaller) than any r.i. space X(Ω) such that $W_0^{k, Y} (\Omega)\subset X (\Omega)$. This result extends, complements, simplifies and sharpens recent results in [10].
- Subjects
EMBEDDING theorems; EMBEDDINGS (Mathematics); ALGEBRAIC geometry; SOBOLEV spaces; REARRANGEMENT invariant spaces; FUNCTION spaces
- Publication
Communications in Contemporary Mathematics, 2004, Vol 6, Issue 3, p495
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199704001380