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 $\|\mathcal{H}\|_{A_\alpha^{2m}\to A_\alpha^{2m}}=B(a,1-a)$, where $a=(\alpha+2)/(2m)$, whenever $0 B(a_0,1-a_0)$. 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 $m\geq 550000$.