Marcinkiewicz <i>r</i>-classes and Fourier series expansions of operator ergodic Stieltjes convolutions

Earl Berkson

doi:10.1016/j.crma.2013.10.017

Harmonic analysis/Functional analysis

Marcinkiewicz r-classes and Fourier series expansions of operator ergodic Stieltjes convolutions

Presented by: Gilles Pisier

Earl Berkson ¹

¹ Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA

Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 813-815

Abstract (Original language)
Résumé

We study the Fourier series expansions in the strong operator topology for operator-valued Stieltjes convolutions of Marcinkiewicz r-classes against spectral decompositions of modulus-mean-bounded operators. The vector-valued harmonic analysis resulting can be viewed as an extension of traditional Calderón–Coifman–G. Weiss transference without being constrained by the latterʼs requirement of power-boundedness.

Cette note étudie (dans la topologie forte des opérateurs) les développements en séries de Fourier pour les « convolutions de Stieltjes » des fonctions dans les r-classes de Marcinkiewicz par $d E$ , où E est la décomposition spectrale dʼune bijection linéaire arbitraire T telle que T soit un opérateur préservant la disjonction dont le module linéaire est à moyennes bornées. Lʼanalyse harmonique vectorielle qui en résulte étend le transfert traditionnel de Calderón–Coifman–G. Weiss, sans supposer les puissances uniformément bornées traditionnellement requises pour le transfert.

Received: 2013-09-25
Accepted: 2013-10-17
Published online: 2013-11-01

DOI: 10.1016/j.crma.2013.10.017

Author's affiliations:

Earl Berkson ¹

¹ Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA

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     title = {Marcinkiewicz \protect\emph{r}-classes and {Fourier} series expansions of operator ergodic {Stieltjes} convolutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {813--815},
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Earl Berkson. Marcinkiewicz r-classes and Fourier series expansions of operator ergodic Stieltjes convolutions. Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 813-815. doi: 10.1016/j.crma.2013.10.017

References
Cited by

[1] E. Berkson Spectral theory and operator ergodic theory on super-reflexive Banach spaces, Stud. Math., Volume 200 (2010), pp. 221-246

[2] E. Berkson Rotation methods in operator ergodic theory, N.Y. J. Math., Volume 17 (2011), pp. 21-39 http://nyjm.albany.edu/j/2011/17-2v.pdf (located online at)

[3] E. Berkson; T.A. Gillespie Mean-boundedness and Littlewood–Paley for separation-preserving operators, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 1169-1189

[4] E. Berkson; T.A. Gillespie Shifts as models for spectral decomposability on Hilbert space, J. Oper. Theory, Volume 50 (2003), pp. 77-106

[5] R. Coifman; J.L. Rubio de Francia; S. Semmes Multiplicateurs de Fourier de $L^{p} (R)$ et estimations quadratiques, C. R. Acad. Sci. Paris, Ser. I, Volume 306 (1988), pp. 351-354

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