Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces

Abstract

If $\,\mu \,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_\mu \,$ be the Hankel matrix $\,(\mu _{n, k})_{n,k\ge 0}\,$ with entries $\,\mu _{n, k}=\mu _{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots $, $\mu_n\,$ denotes the moment of order $\,n\,$ of $\,\mu $. This matrix induces formally the operator $\,\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D $. When $\,\mu \,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_\mu\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1<p<\infty $, but not on $\,H^1$. J. Cima has recently proved that $\,\mathcal H\,$ is an injective bounded operator from $\,H^1\,$ into the space $\,\mathscr C\,$ of Cauchy transforms of measures on the unit circle. \par The operator $\,\mathcal H_\mu \,$ is known to be well defined on $\,H^1\,$ if and only if $\,\mu \,$ is a Carleson measure and in such a case we have that $\mathcal H_\mu (H^1)\subset \,\mathscr C$. Furthermore, it is bounded from $\,H^1\,$ into itself if and only if $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure. \par In this paper we prove that when $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure then $\,\mathcal H_\mu \,$ actually maps $\,H^1\,$ into the space of Dirichlet type $\,\mathcal D^1_0\,$. We discuss also the range of $\,\mathcal H_\mu\,$ on $\,H^1\,$ when $\,\mu \,$ is an $\alpha $-logarithmic $1$-Carleson measure ($0<\alpha <1$). We study also the action of the operators $\,\mathcal H_\mu \,$ on Bergman spaces and on Dirichlet spaces.