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Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces (1804.02227v3)

Published 6 Apr 2018 in math.CV

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 $\,\mathcal{H}\mu (f)(z)= \sum_{n=0}{\infty}\left(\sum_{k=0}{\infty} \mu_{n,k}{a_k}\right)zn\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}\infty a_kzk\,$, 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 $\,Hp\,$ if $\,1<p<\infty $, but not on $\,H1$. J. Cima has recently proved that $\,\mathcal H\,$ is an injective bounded operator from $\,H1\,$ 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 $\,H1\,$ if and only if $\,\mu \,$ is a Carleson measure and in such a case we have that $\mathcal H_\mu (H1)\subset \,\mathscr C$. Furthermore, it is bounded from $\,H1\,$ 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 $\,H1\,$ into the space of Dirichlet type $\,\mathcal D1_0\,$. We discuss also the range of $\,\mathcal H_\mu\,$ on $\,H1\,$ 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.

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