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

If $\mu$ is a finite positive Borel measure on the interval $[0,1)$, we let ${\mathcal{H}}_{\mu }$ be the Hankel matrix $,(\mu {n, k}){n,k\ge 0},$withentries$,\mu {n, k}=\mu {n+k}$,where,for$,n,=,0, 1, 2, \dots $,$\mu_n,$denotesthemomentoforder$,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 $f(z)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{z}^k$, 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 $1<p<\mathrm{\infty }$, but not on $H^1$. J. Cima has recently proved that $\mathcal{H}$ is an injective bounded operator from $H^1$ into the space $\mathcal{C}$ of Cauchy transforms of measures on the unit circle. \par The operator $,\mathcal H_\mu ,$isknowntobewelldefinedon$,H^1,$ifandonlyif$,\mu ,$ is a Carleson measure and in such a case we have that $\mathcal H_\mu (H^1)\subset ,\mathscr C$\mathrm{.}Furthermore,itisboundedfrom$,H^1,$ 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 $,\mathcal H_\mu ,$actuallymaps$,H^1,$intothespaceofDirichlettype$,\mathcal D^1_0,$\mathrm{.}Wediscussalsotherangeof$,\mathcal H_\mu,$on$,H^1,$when$,\mu ,$isan$\alpha $-logarithmic$1$-Carlesonmeasure($0<\alpha <1$). We study also the action of the operators ${\mathcal{H}}_{\mu }$ on Bergman spaces and on Dirichlet spaces.