Strong convergence in the weighted setting of operator-valued Fourier series defined by the Marcinkiewicz multipliers

Earl Berkson

doi:10.1016/j.crma.2015.11.001

Harmonic analysis/Functional analysis

Strong convergence in the weighted setting of operator-valued Fourier series defined by the Marcinkiewicz multipliers

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 354 (2016) no. 2, pp. 181-184.

Abstract
Résumé

Suppose that $1 < p < \infty$ and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt $A_{p}$ weight condition. We show that every Marcinkiewicz multiplier $ψ : T \to C$ has an associated operator-valued Fourier series which serves as an analogue in $B (ℓ^{p} (w))$ of the usual Fourier series of ψ, and this operator-valued Fourier series is everywhere convergent in the strong operator topology. In particular, we deduce that the partial sums of the usual Fourier series of ψ are uniformly bounded in the Banach algebra of Fourier multipliers for $ℓ^{p} (w)$ . These results transfer to the framework of invertible, modulus mean-bounded operators acting on $L^{p}$ spaces of sigma-finite measures.

Soient $1 < p < \infty$ et w un poids dans la classe $A_{p} (Z)$ . Cette note établit (dans la topologie forte des opérateurs) la convergence des séries de Fourier (à valeurs dans $B (ℓ^{p} (w))$ ) pour les « convolutions de Stieltjes », où ces convolutions sont déterminées par les fonctions ψ appartenant à la classe de Marcinkiewicz $M_{1} (T)$ . Les propriétés de convergence pour ces séries de Fourier ayant valeurs dans $B (ℓ^{p} (w))$ révèlent des propriétés de convergence des séries de Fourier traditionnelles pour les fonctions $ψ \in M_{1} (T)$ . En particulier, les sommes partielles de la série de Fourier traditionnelle pour un ψ∈ $M_{1} (T)$ quelconque sont uniformément bornées dans la norme des p-multiplicateurs pour $ℓ^{p} (w)$ . Ces résultats se transfèrent immêdiatement au cadre d'une bijection linéaire arbitraire T telle que T soit un opérateur préservant la disjonction dont le module linéaire est à moyennes bornées.

Received: 2015-07-04
Accepted: 2015-11-04
Published online: 2016-01-11

DOI: 10.1016/j.crma.2015.11.001

Keywords: Marcinkiewicz class, Fourier multiplier, Fourier series, Modulus mean-bounded operator, Shift operator, $ {A}_{p}$ weight sequence

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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     author = {Earl Berkson},
     title = {Strong convergence in the weighted setting of operator-valued {Fourier} series defined by the {Marcinkiewicz} multipliers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {181--184},
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Earl Berkson. Strong convergence in the weighted setting of operator-valued Fourier series defined by the Marcinkiewicz multipliers. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 181-184. doi : 10.1016/j.crma.2015.11.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.001/

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 Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series, Stud. Math., Volume 222 (2014), pp. 123-155

[4] J. Duoandikoetxea Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal., Volume 260 (2011), pp. 1886-1901

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