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 orden $n$ 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 $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_\mu $ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.Comment: 24 page

If µ is a positive Borel measure on the interval [0, 1) we let H µ be the Hankel matrix H µ = (µ n,k ) n,k≥0 with entries µ n,k = µ n+k , where, for n = 0, 1, 2, . . . , µ n denotes the moment of order n of µ. This matrix induces formally the operatorµ n,k a k z n on the space of all analytic functions f (z) = ∞ k=0 a k z k , in the unit disc D. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators H µ on Hardy spaces and on Möbius invariant spaces.

Let $${\mathbb {D}}$$ D be the unit disc in $${\mathbb {C}}$$ C . If $$\mu $$ μ is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in $${\mathbb {D}}$$ D , $$f(z)=\sum _{n=0}^\infty a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n ($$z\in {\mathbb {D}}$$ z ∈ D ), we define $$\begin{aligned} {\mathcal {C}}_\mu (f)(z)= \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n,\quad z\in {\mathbb {D}}, \end{aligned}$$ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where, for $$n\ge 0$$ n ≥ 0 , $$\mu _n$$ μ n denotes the n-th moment of the measure $$\mu $$ μ , that is, $$\mu _n=\int _{[0, 1)}t^nd\mu (t).$$ μ n = ∫ [ 0 , 1 ) t n d μ ( t ) . In this way, $${\mathcal {C}}_\mu $$ C μ becomes a linear operator defined on the space $${\mathrm{Hol}}({\mathbb {D}})$$ Hol ( D ) of all analytic functions in $${\mathbb {D}}$$ D . We study the action of the operators $${\mathcal {C}}_\mu $$ C μ on distinct spaces of analytic functions in $${\mathbb {D}}$$ D , such as the Hardy spaces $$H^p$$ H p , the weighted Bergman spaces $$A^p_\alpha $$ A α p , BMOA, and the Bloch space $${\mathcal {B}}$$ B .

To each weighted Dirichlet space D p , 0 < p < 1, we associate a family of Morrey-type spaces D λ p , 0 < λ < 1, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.Recall that the space BMOA consists of all functions f ∈ H 2 whose boundary values f (e iθ ) have bounded mean oscillation, that is,

If µ is a positive Borel measure on the interval [0, 1) we let Hµ be the Hankel matrix Hµ = (µ n,k ) n,k≥0 with entries µ n,k = µ n+k , where, for n = 0, 1, 2, . . . , µn denotes the moment of order n of µ. This matrix induces formally the operator 1