A derivative-Hilbert operator acting from logarithmic Bloch spaces to Bergman spaces

Abstract

Let μ be a positive Borel measure on the interval [0, 1). The Hankel matrix \(\cal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}\) with entries μ_n,k = μ_n+k, where μ_n = ⨜_[0,1) tⁿdμ(t), induces, formally, the operator

$$\cal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\left(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n, ~z\in \mathbb{D},$$

where \(f(z)=\sum\limits_{n=0}^\infty a_nz^n\) is an analytic function in ⅅ. We characterize the measures μ for which \(\cal{DH}_\mu\) is bounded (resp., compact) operator from the logarithmic Bloch space \(\mathscr{B}_{L^{\alpha}}\) into the Bergman space \(\cal{A}^p\), where 0 ≤ α < ∞, 0 < p < ∞. We also characterize the measures μ for which \(\cal{DH}_\mu\) is bounded (resp., compact) operator from the logarithmic Bloch space \(\mathscr{B}_{L^{\alpha}}\) into the classical Bloch space \(\mathscr{B}\).