A variant of Hilbert's inequality and the norm of the Hilbert Matrix on K^p

2023/07/19 by Vassilis Daskalogiannis, Daskalogiannis, Vassilis, Πέτρος Γαλανόπουλος +3
Mathematics · #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2307.09859

Abstract

We prove the nontrivial variant ∑m,n=1∞((n)/(m))(1)/(q)-(1)/(p)(ambn)/(m+n-1)≤\fracπsin\fracπp ( ∑m=1∞amp)\frac 1p( ∑n=1∞bnq)\frac 1q of the well known Hilbert's inequality. Then we use this to determine the exact value \fracπsin\fracπp of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space Kp. This space consists of all functions f(z)=∑m=0∞amzm analytic in the unit disc with ‖f‖Kpp=∑m=0∞(m+1)p-2|am|p<+∞.

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