Function spaces of Lizorkin–Triebel type on an irregular domain

✍ Scribed by O.V. Besov

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 309 KB
• Volume: 70
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2008.12.038

✦ Synopsis

In 1938 S.L. Sobolev proved his well-known embedding theorem

for domains G ⊂ R n satisfying the cone condition (see [1]). Relation (2) (which determines the maximum possible value of q in theorem ( 1)) is also a necessary condition for the embedding. Sobolev's result has been extended to domains of a more general form: domains of the classes J n-1 n and I p, Maz'ya, 1960Maz'ya, , 1975, see [2], see [2]), John domains (Yu.G. Reshetnyak [3,4]), and domains with the flexible cone condition (O.V. Besov, 1983, see [5]).

Definition 1 ([6]). For σ ≥ 1 a domain G ⊂ R n is said to satisfy the flexible σ -cone condition if, for some T > 0 and 0 < κ 0 ≤ 1, for each x ∈ G there exists a piecewise smooth path