Mean Lipschitz spaces and a generalized Hilbert operator

If $\mu$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu$ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ 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 $\mathbb{D}$. This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $\mathcal H_\mu$ on mean Lipschitz spaces of analytic functions.