Hankel matrices acting on the Hardy space $$H^1$$ 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 $$\,n\,=\,0, 1, 2, \ldots $$ , $$\mu _n\,$$ 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. 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. 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.