Banach spaces related to integrable group representations and their atomic decompositions. II. (English) Zbl 0713.43004
Summary: We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact group \({\mathcal G}\) and every reasonable function space on \({\mathcal G}\). Whereas Part I [see the authors, J. Funct. Anal. 86, 307–340 (1989; Zbl 0691.46011)] was devoted to atomic decompositions of such spaces Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples [Part III (to appear)] one recovers several and often difficult theorems with ease.
MSC:
| 43A99 | Abstract harmonic analysis |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
Keywords:
coorbit spaces; integrable, irreducible, unitary representation; locally compact group; atomic decompositionsCitations:
Zbl 0691.46011References:
| [1] | [BL]Bergh-Löfström: Interpolation Spaces (An Introduction). Berlin-Heidelberg-New York: Springer. 1976. · Zbl 0344.46071 |
| [2] | [F1]Feichtinger, H. G.: Banach convolution algebras of Wiener’s type. Proc. Conf. ?Functions, Series, Operators? Budapest, August 1980. Colloquia Math. Soc. J. Bolyai. Amsterdam-Oxford-New York: North-Holland. 1983, 509-524. |
| [3] | [BF]Feichtinger, H. G., Braun, W.: Banach spaces of distributions having two module structures. J. Funct. Analysis51, 174-212 (1983). · Zbl 0515.46045 · doi:10.1016/0022-1236(83)90025-3 |
| [4] | [LRP]Feichtinger, H. G., Gröchenig, K. H.: A unified approach to atomic decompositions via integrable group representations. Lect. Notes Math.1302, 52-73. Berlin-Heidelberg-New York: Springer. 1988. · Zbl 0658.22007 |
| [5] | [FG]Feichtinger, H. G., Gröchenig, K. H.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. To appear. (1989). · Zbl 0713.43004 |
| [6] | [GR]Gröchenig, K.H.: Unconditional bases in translation and dilation invariant function spaces onR n. Sc. Proc. Conf. on Constructive Theory of Functions. Varna 1987 (Bulgaria), pp. 174-183 (eds.:B. Sandov et al.). Sofia: Bulgar. Acad. Sci. 1988. · Zbl 0726.42015 |
| [7] | [LT]Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. Berlin-Heidelberg-New York: Springer. 1977. · Zbl 0362.46013 |
| [8] | [M]Meyer, Y.: Principe d’incertitude, bases hilbertiennes et algébres d’operateurs. Sèminaire Bourbaki, 38ème année, 1985/86, n0 662. |
| [9] | [P]Pietsch, A.: Operator Ideals. Berlin: VEB Deutscher Verlag der Wiss. 1978; Amsterdam-New York-Oxford: North-Holland. 1980. |
| [10] | [TR]Triebel, H.: Theory of Function Spaces. Leipzig: Akad. Verlagsges. 1983. · Zbl 0546.46028 |
| [11] | [Z]Zaanen, A.: Integration. Amsterdam: North-Holland. 2nd ed. 1967. |
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