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Estimates of \(L_ p\)-moduli of continuity and imbedding theorems for a domain with the flexible horn condition. (English. Russian original) Zbl 0595.46030

Sov. Math., Dokl. 29, 329-333 (1984); translation from Dokl. Akad. Nauk SSSR 275, 1036-1041 (1984).
Given \(\lambda =(\lambda_ 1,..,\lambda_ n)\in (0,\infty)^ n\), a domain \(G\subset R^ n\) is said to be a domain with the flexible \(\lambda\)-horn condition (the flexible cone condition when \(\lambda_ 1=...=\lambda_ n)\) if for some \(\delta_ 0\in (0,1]\), \(T\in (0,\infty)\) and for any \(x\in G\), there exists a curve in \(R^ n\) \(\rho (t^{\lambda})=^{def}(\rho_ 1(t^{\lambda_ 1}),...,\rho_ n(t^{\lambda_ n}))=\rho (t^{\quad \lambda},x)\), \(0\leq t\leq T\), with the following properties:
a) \(\rho_ i(u)\) is absolutely continuous and \(| \rho_ i'(u)| \leq 1\) a.e. on \([0,T^{\lambda_ i}]\), \(i=1,...,n,\)
b) \(\rho (0)=0\) and \(V(\lambda,x,\delta_ 0)=^{def}x+\cup_{0<t\leq T}[\rho (t^{\lambda})+t^{\lambda}\delta_ 0\) \(^{\lambda}Q_ 0]\subset G\), where we use the notation \(t^{\lambda}=(t^{\lambda_ 1},...,t^{\lambda_ n})\), \(xy=(x_ 1y_ 1,...,x_ ny_ n)\) and \(Q_ 0=[-1,1]^ n\). The set \(V(\lambda,x,\delta_ 0)\) is called a flexible \(\lambda\)-horn (with the vertex at x). With respect to an ordinary horn: \(\rho (t^{\lambda})=at^{\lambda}\), \(a\in R^ n\setminus \{0\}\), integral representations of a function of several variables in terms of its derivatives or in terms of its differences of various orders and their derivatives have first been constructed and then developed by S. L. Sobolev, O. V. Besov, V. P. Il’in and S. M. Nikol’skij. In this note such are generalized to a domain with the flexible \(\lambda\)-horn condition and various estimates of \(L_ p\)- moduli of continuity and some imbedding theorems are obtained. The present reviewer, however, has much difficulty in giving a short account of particulars.
Reviewer: K.Yoshinaga

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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