Unions et intersections d’espaces $L^p$ invariantes par translation ou convolution

Bertrandias, Jean-Paul ; Datry, Christian ; Dupuis, Christian

Annales de l'Institut Fourier, Tome 28 (1978), p. 53-84 / Harvested from Numdam

Résumé

Étude des propriétés des unions et intersections d’espaces $L^p(s)$ relatifs à un ensemble $S$ de mesures positives sur un groupe commutatif localement compact lorsque $S$ est invariant par translation ou stable par convolution.

Dans des cas particuliers, on retrouve les propriétés d’espaces étudiés par A. Beurling et par B. Koremblium.

On étudie aussi les espaces $\ell ^p(L^{p^{\prime }})$ formés des fonctions appartenant localement à $L^{p^{\prime }}$ et qui ont un comportement $\ell ^p$ à l’infini.

Abstract

This paper is concerned with properties of unions and intersections of $L^p(s)$ spaces where $s$ belongs to a set $S$ of positive measures on a locally compact abelian group and where $S$ is translation invariant or convolution invariant.

In special cases, we find again properties of spaces studied by A. Beurling and by B. Koremblium.

We also study the spaces $\ell ^p(L^{p^{\prime }})$ of functions belonging locally to $L^{p^{\prime }}$ and with $\ell ^p$ behaviour at infinity.

Détail

Publié le : 1978-01-01

MR 81g:43005 | Zbl 0365.46029 | 3 citations dans Numdam

DOI : https://doi.org/10.5802/aif.689

Bibliographie

[1] L. N. Argabright and J. Gil De Lamadrid, Fourier analysis of unbounded measures on locally compact abelian groups, Memoirs of the Ann. Math. Soc., n° 145 (1974). | MR 58 #29842 | Zbl 0294.43002

[2] A. Benedek and R. Panzone, The spaces Lp, with mixed norm, Duke Math. J., 28 (1961), 301-324. | Zbl 0107.08902

[3] C. Berg and G. Forst, Potential theory on locally compact abelian groups, Springer 1975. | MR 58 #1204 | Zbl 0308.31001

[4] J.-P. Bertrandias, Unions et intersections d'espaces Lp sur un espace localement compact, Bull. Sc. Math., 101 (1977). | MR 58 #23529 | Zbl 0376.46016

[5] J.-P. Bertrandias et C. Dupuis, Transformation de Fourier sur les espace lp(Lp'), Ann. Inst. Fourier 29 (1979). | Numdam | MR 82a:43006 | Zbl 0387.42003

[6] A. Beurling, Construction and analysis of some convolution algebras, Ann. Inst. Fourier, 14 (1964), 1-32. | Numdam | MR 32 #321 | Zbl 0133.07501

[7] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge, 1974. | MR 56 #12824 | Zbl 0273.46035

[8] R. Goldberg, On a space of functions of Wiener, Duke Math. J., 34 (1967), 683-691. | MR 36 #703 | Zbl 0172.16203

[9] E. Hewitt, A survey of abstract harmonic analysis, dans : Some aspects of analysis and probability, Wiley & sons, 1958. | MR 21 #2159 | Zbl 0089.10806

[10] E. Hewitt and K. A. Ross, Abstract harmonic analysis. 2 volumes, Springer, 1963 et 1970.

[11] F. Holland, Harmonic analysis on amalgams of Lp and Lq, J. London Math. Soc., 2, 10 (1975), 295-305. | MR 51 #11013 | Zbl 0314.46029

[12] B. Koremblium, On certain special commutative normed rings. (en russe), Doklady Akad. Nauk SSSR, 64 (1949), 281-284.

[13] R. Larsen, Measures with separable orbits, Proc. Am. Math. Soc., 19 (1968), 569-572. | MR 37 #6689 | Zbl 0157.06801

[14] H. Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. | MR 46 #5933 | Zbl 0165.15601

[15] H. Reiter, L1-algebras and Segal algebras, Lectures Notes n° 231, Springer 1971. | MR 55 #13158 | Zbl 0219.43003

[16] W. Rudin, Measures algebras on abelian groups, Bull. Am. Math. Soc., 65 (1959), 227-247. | MR 21 #7404 | Zbl 0089.10901

[17] P. Szeptycki, On functions and measures whose Fourier transforms are functions, Math. Ann., 179 (1968), 31-41. | MR 39 #710 | Zbl 0167.41601

[18] N. Wiener, Tauberian theorems, Ann. of Math., 33 (1932), 1-100. | JFM 58.0226.02 | Zbl 0004.05905