## 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
Full Text:

### References:

 [1] Aubin, T., Espaces de Sobolev sur les variétés Riemanniennes,Bull. Sci. Math. 100 (1976), 149–173. · Zbl 0328.46030 [2] Aubin, T.,Nonlinear analysis on manifolds. Monge–Ampère equations, Springer, New York, Heidelberg, Berlin, 1982. · Zbl 0512.53044 [3] Bergh, J. andLöfström, J.,Interpolation spaces. An introduction, Springer, Berlin, Heidelberg, New York, 1976. · Zbl 0344.46071 [4] Choquet-Bruhat, Y. andChristodoulou, D., Elliptic systems inH s {$$\delta$$} spaces on manifolds which are Euclidean at infinity,Acta Math. 146 (1981), 129–150. · Zbl 0484.58028 [5] Colzani, L.,Lipschitz spaces on compact rank one symmetric spaces. In: Harmonic analysis, Proceedings, Cortona 1982. Lecture Notes Math. 992, 139–160. Springer, Berlin, Heidelberg, New York, Tokyo, 1983. [6] Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups,Ark. Mat. 13 (1975), 161–207. · Zbl 0312.35026 [7] Folland, G. B., Lipschitz classes and Poisson integrals on stratified groups,Studia Math. 66 (1979), 37–55. · Zbl 0439.43005 [8] Folland, G. B. andStein, E. M.,Hardy spaces on homogeneous groups, Princeton Univ. Press, Princeton, 1982. [9] Geisler, M., Funktionenräume auf kompakten Lie-Gruppen, Dissertation, Jena, 1984. [10] Geisler, M., Function spaces on compact Lie groups,Math. Nachr. · Zbl 0703.43007 [11] Helgason, S.,Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, San Francisco, London, 1978. · Zbl 0451.53038 [12] Klingenberg, W.,Riemannian geometry, W. de Gruyter, Berlin, New York, 1982. · Zbl 0495.53036 [13] Krantz, S. G., Lipschitz spaces on stratified groups,Trans. Amer. Math. Soc. 269 (1982), 39–66. · Zbl 0529.22006 [14] Lizorkin, P. I., Properties of functions of the spaces {$$\Lambda$$} p{$$\theta$$} r (Russian),Trudy Mat. Inst. Steklov 131 (1974), 158–181. [15] Peetre, J., Sur les espaces de Besov,C. R. Acad. Sci. Paris, Sér A–B 264 (1967), 281–283. · Zbl 0145.16206 [16] Peetre, J., Remarques sur les espaces de Besov. Le cas 0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.