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

Abstract

We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\fracπ{\sin\fracπ{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big( \sum\limits_{n=1}^{\infty}b_n^q\Big)^{\frac 1q} \] 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 $f(z)=\sum _{m=0}^{\infty }{a}_m{z}^m$ analytic in the unit disc with $\Vert f{\Vert }_{K^p}^p=\sum _{m=0}^{\infty }(m+1{)}^{p-2}|a_m{|}^p<+\infty$.

14 pages