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Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds. (English) Zbl 0664.46026

The families of spaces \(B^ s_{pq}\) and \(F^ s_{pq}\) \((-\infty <s<\infty,0<p\leq \infty)\) were introduced by J. Peetre, P. I. Lizorkin, and the author. For a suitable choice of parameters these families include most of the known function spaces defined by differentiability and smoothness conditions. In the present article these spaces are defined on a complete Riemannian manifold that satisfies suitable uniformity conditions. The project requires a local definition of the spaces in question which is to be found in previous work of the author. It turns out that while there is a natural definition of the F spaces on a manifold, some technical difficulties arise in the case of B spaces; in the present paper these are defined by means of real interpolation. Various results about spaces on manifolds are established, in particular, inclusions and isomorphisms by means of Bessel kernels and interpolation properties. Forthcoming papers on the subject are announced.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58D15 Manifolds of mappings
46M35 Abstract interpolation of topological vector spaces
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