Hilbert matrix norms on weighted Bergman spaces: even exponents and a counterexample to the beta formula

Abstract

Let $A_{\alpha }^p$ be the weighted Bergman space on the unit disk, where $\alpha >-1$. For $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k\in A_{\alpha }^p$, consider the Hilbert matrix operator $\mathcal{H}f(z)={\sum }_{n=0}^{\infty }\left({\sum }_{k=0}^{\infty }\frac{a_k}{n+k+1}\right)z^n={\int }_0^1\frac{f(t)}{1-tz}dt$. For even exponents $p=2m$, we prove that $\Vert \mathcal{H}{\Vert }_{A_{\alpha }^{2m}\to A_{\alpha }^{2m}}=B(a,1-a)$, where $a=(\alpha +2)/(2m)$, whenever $0<a\le m/(2m-1)$. For $p=2,4,6,8,10$, the same formula holds throughout the admissible range. We also show that the beta-function norm formula does not hold for all admissible parameters. Set $a_0=800001/1000000$ and ${\alpha }_p=a_{0}p-2$. Then, for every real $p\ge 1100000$, $\Vert \mathcal{H}{\Vert }_{A_{{\alpha }_p}^p\to A_{{\alpha }_p}^p}>B(a_0,1-a_0)$. The counterexample is based on the fixed function $f_0(z)=(1-z^2{)}^{-4/5}={\sum }_{k=0}^{\infty }\frac{(4/5{)}_k}{k!}{z}^{2k}$. A rigorous interval estimate at $p=1100000$, together with monotonicity in $p$, yields the result on the entire half-line. In particular, the formula fails for every even exponent $p=2m$ with $m\ge 550000$.

22 pages. We disprove Karapetrović's conjectured beta-function formula for the norm of the Hilbert matrix operator on weighted Bergman spaces