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

We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\frac{\pi}{\sin\frac{\pi}{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big( \sum\limits_{n=1}^{\infty}b_n^q\Big)^{\frac 1q} \] of the well known Hilbert&...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Authors: Daskalogiannis, Vassilis, Galanopoulos, Petros, Papadimitrakis, Michael
Format: Article
Language:English
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Summary:We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\frac{\pi}{\sin\frac{\pi}{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big( \sum\limits_{n=1}^{\infty}b_n^q\Big)^{\frac 1q} \] of the well known Hilbert's inequality. Then we use this to determine the exact value \(\frac{\pi}{\sin\frac{\pi}p}\) of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space \(K^p\). This space consists of all functions \(f(z)=\sum\limits_{m=0}^{\infty}a_mz^m\) analytic in the unit disc with \(\|f\|_{K^p}^p=\sum\limits_{m=0}^{\infty}(m+1)^{p-2}|a_m|^p
ISSN:2331-8422