Continuous generalization of Clarkson-McCarthy inequalities

2018/01/06 by Kečkić, Dragoljub J.
#43A25 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47A30 #Secondary: 47B10

paper · doi:10.48550/arxiv.1801.02103

Abstract

Let G be a compact abelian group, let μ be the corresponding Haar measure, and let G be the Pontryagin dual of G. Further, let C_p denote the Schatten class of operators on some separable infinite dimensional Hilbert space, and let L^p(G;C_p) denote the corresponding Bochner space. If G\niθ↦ A_θ is the mapping belonging to L^p(G;C_p) then, ∑_{k∈ G}‖∫_Gk(θ)A_θ dθ‖_p^p≤∫_G‖A_θ‖_p^p dθ, p≥2 ∑_{k∈ G}‖∫_Gk(θ)A_θ dθ‖_p^p≤(∫_G‖A_θ‖_p^q dθ)^p/q, p≥2. ∑_{k∈ G}‖∫_Gk(θ)A_θ dθ‖_p^q≤(∫_G‖A_θ‖_p^p dθ)^q/p, p≤2. If G is a finite group, the previous comprises several earlier obtained generalizations of Clarkson-McCarthy inequalities (e.g. G=Z_n or G=Z₂ⁿ), as well as the original inequalities, for G=Z₂. Other related inequalities are also obtained.

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