Banach Journal of Mathematical Analysis

Continuous generalization of Clarkson–McCarthy inequalities

Dragoljub J. Kečkić

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Abstract

Let G be a compact Abelian group, let μ be the corresponding Haar measure, and let Gˆ be the Pontryagin dual of G. Furthermore, let Cp denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let Lp(G;Cp) denote the corresponding Bochner space. If GθAθ is the mapping belonging to Lp(G;Cp), then kGˆGk(θ)¯AθdθppGAθppdθ,p2,kGˆGk(θ)¯Aθdθpp(GAθpqdθ)p/q,p2,kGˆGk(θ)¯Aθdθpq(GAθppdθ)q/p,p2. If G is a finite group, then the previous equations comprise several generalizations of Clarkson–McCarthy inequalities obtained earlier (e.g., G=Zn or G=Z2n), as well as the original inequalities, for G=Z2. We also obtain other related inequalities.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 26-46.

Dates
Received: 17 January 2018
Accepted: 19 April 2018
First available in Project Euclid: 28 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1538121808

Digital Object Identifier
doi:10.1215/17358787-2018-0014

Mathematical Reviews number (MathSciNet)
MR3894063

Zentralblatt MATH identifier
07002030

Subjects
Primary: 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Keywords
Clarkson inequalities unitarily invariant norm abstract Fourier series finite group Littlewood matrices

Citation

Kečkić, Dragoljub J. Continuous generalization of Clarkson–McCarthy inequalities. Banach J. Math. Anal. 13 (2019), no. 1, 26--46. doi:10.1215/17358787-2018-0014. https://projecteuclid.org/euclid.bjma/1538121808


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