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**Einige neuere Entwicklungen in der Theorie der Funktionenräume. (Some new developments in the theory of function spaces).**
*(German)*
Zbl 0638.46025

This note is a historical summary of the theory of function spaces. From the year 1935 to 1938 S. L. Sobolev has published three papers and has developed his function spaces \(W\) \(k_ p\) which are called Sobolev spaces. Also, there are many investigations of function spaces such as homogeneous spaces, anisotropic spaces, spaces of Sobolev-Orlicz type which are different from \(W\) \(k_ p\). Around 1960, it was established the theory of Hölder-Zygmund spaces, Bessel-Lipschitz spaces, Hardy spaces and BMO spaces. It is asked to find a systematic principle to investigations of function spaces. The first results of this direction are succeeded by help of abstract interpolation theory (real and complex method). By other powerful tools, it was established the theory of \(B\) \(s_{p,q}\) or \(F\) \(s_{p,q}\) around 1970 to 1975.

The author of this note delivered the history of the theory with definitions and properties. Titles of the note are as follows:

1. Introduction.

2. Constructive methods. 2.1 Hölder-Zygmund spaces, 2.2 Sobolev spaces, 2.3 Besov-Lipschitz spaces, 2.4 Bessel-Potential spaces, 2.5 Properties of spaces C s, \(A\) \(s_{p,q}\) and \(H\) \(s_ p\), 2.6 Hardy spaces, BMO spaces.

3. Methods of interpolation. 3.1 Interpolation of function spaces, 3.2 real interpolation.

4. The Fourier-analytic methods. 4.1 Definitions, 4.2 Properties, 4.3 equivalent quasi-norms, 4.4 equivalent quasi-norms (mean-value of differentiation), 4.5 interpolation.

5. Local or global methods. 5.1 equivalent quasi-norms (general cases), 5.2 Harmonic and heat extensions, 5.3 Local-global characterizations.

6. Spaces on Riemannian manifolds. 6.1 Complete Riemannian manifolds with bounded geometry, 6.2 special function spaces on manifolds, 6.3 Definitions, 6.4 Properties, 6.5 inner descriptions, 6.6 equivalent quasi-norms (difference), 6.7 equivalent quasi-norms (mean-value of differentiation).

7. Spaces on Lie groups. 7.1 Riemannian entrance, 7.2 compact Lie group, 7.3 general Lie group.

This note is a summary of his lecture talk in XI Österreichischer Mathematikkongress (Austrian Mathematical Congress) in Graz at 16-20, September 1985.

The author of this note delivered the history of the theory with definitions and properties. Titles of the note are as follows:

1. Introduction.

2. Constructive methods. 2.1 Hölder-Zygmund spaces, 2.2 Sobolev spaces, 2.3 Besov-Lipschitz spaces, 2.4 Bessel-Potential spaces, 2.5 Properties of spaces C s, \(A\) \(s_{p,q}\) and \(H\) \(s_ p\), 2.6 Hardy spaces, BMO spaces.

3. Methods of interpolation. 3.1 Interpolation of function spaces, 3.2 real interpolation.

4. The Fourier-analytic methods. 4.1 Definitions, 4.2 Properties, 4.3 equivalent quasi-norms, 4.4 equivalent quasi-norms (mean-value of differentiation), 4.5 interpolation.

5. Local or global methods. 5.1 equivalent quasi-norms (general cases), 5.2 Harmonic and heat extensions, 5.3 Local-global characterizations.

6. Spaces on Riemannian manifolds. 6.1 Complete Riemannian manifolds with bounded geometry, 6.2 special function spaces on manifolds, 6.3 Definitions, 6.4 Properties, 6.5 inner descriptions, 6.6 equivalent quasi-norms (difference), 6.7 equivalent quasi-norms (mean-value of differentiation).

7. Spaces on Lie groups. 7.1 Riemannian entrance, 7.2 compact Lie group, 7.3 general Lie group.

This note is a summary of his lecture talk in XI Österreichischer Mathematikkongress (Austrian Mathematical Congress) in Graz at 16-20, September 1985.

Reviewer: S.Koshi

### MSC:

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

46M35 | Abstract interpolation of topological vector spaces |

46F12 | Integral transforms in distribution spaces |