Norm of the generalized Hilbert operator on Hardy spaces

Abstract

We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\le p\le \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty$ and, by continuous extension to $b=0$, recover the classical norm $\pi /\sin (\pi /p)$. We also prove that ${\mathcal{H}}_b$ is bounded on $H^1$ for every $b>0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]