关键词: derivative-Hilbert operator, logarithmic Bloch space, Carleson measure

Abstract: Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^n{\rm d}\mu(t)$, induces, formally, the operator $\mathcal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\left(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , z\in \mathbb{D},$ where $f(z)=\sum\limits_{n=0}^\infty a_nz^n$ is an analytic function in $\mathbb{D}$. We characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is bounded (resp., compact) operator from the logarithmic Bloch space $\mathscr{B}_{L^{\alpha}}$ into the Bergman space $\mathcal{A}^p$, where $0\leq\alpha<\infty,0<p<\infty$. We also characterize the measures $\mu$ for which $\mathcal{DH}_\mu$ is bounded (resp., compact) operator from the logarithmic Bloch space $\mathscr{B}_{L^{\alpha}}$ into the classical Bloch space $\mathscr{B}$.

Key words: derivative-Hilbert operator, logarithmic Bloch space, Carleson measure