An $L^p$-version of a theorem of D.A. Raikov

Fendler, Gero

doi:10.5802/aif.1002

L^{p}

-version of a theorem of D.A. Raikov

Fendler, Gero

Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 125-135.

Résumé
Abstract

Soit $G$ un groupe localement compact, pour $p \in (1, \infty)$ , soit $P f_{f} (G)$ l’adhérence de $L^{1} (G)$ dans les opérateurs de convolution de $L^{p} (G)$ . Désignons par $W_{p} (G)$ le dual de $P f_{p} (G)$ qui est contenu dans l’espace des multiplicateurs ponctuels de l’espace de Figà-Talamanca Herz $A_{p} (G)$ . On démontre que sur la sphère unité de $W_{p} (G)$ , la topologie $σ (W_{p}, P f_{p})$ et la topologie forte, comme multiplicateurs de $A_{p} (G)$ , coïncident.

Let $G$ be a locally compact group, for $p \in (1, \infty)$ let $P f_{p} (G)$ denote the closure of $L^{1} (G)$ in the convolution operators on $L^{p} (G)$ . Denote $W_{p} (G)$ the dual of $P f_{p} (G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_{p} (G)$ . It is shown that on the unit sphere of $W_{p} (G)$ the $σ (W_{p}, P f_{p})$ topology and the strong $A_{p}$ -multiplier topology coincide.

MR Zbl

DOI : 10.5802/aif.1002

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     author = {Fendler, Gero},
     title = {An $L^p$-version of a theorem of {D.A.} {Raikov}},
     journal = {Annales de l'Institut Fourier},
     pages = {125--135},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {35},
     number = {1},
     year = {1985},
     doi = {10.5802/aif.1002},
     mrnumber = {86h:43003},
     zbl = {0543.43003},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1002/}
}

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Fendler, Gero. An $L^p$-version of a theorem of D.A. Raikov. Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 125-135. doi : 10.5802/aif.1002. http://www.numdam.org/articles/10.5802/aif.1002/

Bibliographie
Cité par

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

[2] F.F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Notes Series 2, Cambridge 1971. | MR | Zbl

[3] M. Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann., 241 (1979), 83-69. | EuDML | MR | Zbl

[4] M. Cowling and G. Fendler, On representations in Banach spaces, Math. Ann., 266 (1984), 307-315. | EuDML | MR | Zbl

[5] P. Eymard, Algèbres Ap et convoluteurs de Lp, Séminaire Bourbaki 22è année, 1969/1970, no. 367. | EuDML | Numdam | Zbl

[6] E.E. Granirer, An application of the Radon Nikodym property in harmonic analysis, Bollentino U.M.I., (5) 18-B (1981), 663-671. | MR | Zbl

[7] E.E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G), Rocky Moutain J. of Math., 11 (1981), 459-472. | MR | Zbl

[8] C. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier, Grenoble, 24-3 (1974), 145-157. | EuDML | Numdam | MR | Zbl

[9] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier Grenoble, 23-3 (1973), 91-123. | EuDML | Numdam | MR | Zbl

[10] G.C. Rota, An “alternierende Verfahen” for general positive operators, Bull. A.M.S., 68 (1962), 95-102. | MR | Zbl

[11] E.M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theorem, Princeton University Press, 1970. | Zbl

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