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

TL;DR

This paper proves a new variant of Hilbert's inequality, then uses it to precisely determine the Hilbert matrix's operator norm on the Hardy-Littlewood space, advancing understanding of operator bounds in complex analysis.

Contribution

It introduces a nontrivial variant of Hilbert's inequality and applies it to exactly compute the Hilbert matrix's norm on the Hardy-Littlewood space.

Findings

Established a new inequality variant with explicit bounds.

Calculated the exact operator norm of the Hilbert matrix on $K^p$.

Enhanced understanding of operator behavior in analytic function spaces.

Abstract

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 $\|f\|_{K^p}^p=\sum\limits_{m=0}^{\infty}(m+1)^{p-2}|a_m|^p<+\infty$.