A derivative-Hilbert operator acting on Dirichlet spaces

Abstract

Let μ\mu be a positive Borel measure on the interval [0,1)\left[0,1). The Hankel matrix Hμ=(μn,k)n,k≥0{{\mathcal{ {\mathcal H} }}}_{\mu }={\left({\mu }_{n,k})}_{n,k\ge 0} with entries μn,k=μn+k{\mu }_{n,k}={\mu }_{n+k}, where μn=∫[0,1)tndμ(t){\mu }_{n}={\int }_{\left[0,1)}{t}^{n}{\rm{d}}\mu \left(t), induces formally the operator as follows: DHμ(f)(z)=∑n=0∞∑k=0∞μn,kak(n+1)zn,z∈D,{{\mathcal{D {\mathcal H} }}}_{\mu }(f)\left(z)=\mathop{\sum }\limits_{n=0}^{\infty }\left(\mathop{\sum }\limits_{k=0}^{\infty }{\mu }_{n,k}{a}_{k}\right)\left(n+1){z}^{n},\hspace{1em}z\in {\mathbb{D}}, where f(z)=∑n=0∞anznf\left(z)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n} is an analytic function in D{\mathbb{D}}. In this article, we characterize those positive Borel measures on [0,1)\left[0,1) for which DHμ{{\mathcal{D {\mathcal H} }}}_{\mu } is bounded (resp. compact) from Dirichlet spaces Dα(0<α≤2){{\mathcal{D}}}_{\alpha }\hspace{0.33em}\left(0\lt \alpha \le 2) into Dβ(2≤β<4){{\mathcal{D}}}_{\beta }\hspace{0.33em}\left(2\le \beta \lt 4).

Keywords

• hilbert operator
• dirichlet space
• carleson measure
• 47b38
• 47b35
• 30h99