This article is cited in 17 papers

Lack of interpolation of linear operators in spaces of smooth functions

B. S. Mityagin, E. M. Semenov


Abstract: We prove that $C^k(\Omega)$, the space of $k$ times continuously differentiable functions on the closure of a region in a finite-dimensional manifold, is not an interpolation space between $C(\Omega)$ and $C^n(\Omega)$ for $0<k<n$. We find analogous results for the Sobolev–Stein spaces. In the class of spaces $C_\varphi$, defined by the modulus of continuity, we describe all interpolation spaces between $C$ and $C^2$.
Bibliography: 34 titles.

UDC: 513.88

MSC: Primary 43A15, 46E15, 46E35; Secondary 26A15, 32E10, 47G05

Received: 13.07.1976

 English version: Mathematics of the USSR-Izvestiya, 1977, 11:6, 1229–1266

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