Continuous generalization of Clarkson-McCarthy inequalities

Original · EN

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 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 Gθ Aθ is the mapping belonging to Lᵖ(G;Cₚ) then, ∑k∈ G∫Gk(θ)Aθdθₚᵖ≤∫GAθₚᵖdθ, p≥2 ∑k∈ G∫Gk(θ)Aθdθₚᵖ≤(∫GAθₚqdθ)ᵖ/q, p≥2. ∑k∈ G∫Gk(θ)Aθdθₚq≤(∫GAθₚᵖdθ)q/ᵖ, p≤2. If G is a finite group, the previous comprises several earlier obtained generalizations of Clarkson-McCarthy inequalities (e.g. G=Zₙ or G=Z₂ⁿ), as well as the original inequalities, for G=Z₂. Other related inequalities are also obtained.