Banach Journal of Mathematical Analysis

Continuous generalization of Clarkson–McCarthy inequalities

Dragoljub J. Kečkić

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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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