Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Let $f$ be an analytic function on the unit disk $\mathbb {D}$. We define a generalized Hilbert-type operator $\mathcal {H}_{a,b}$ by $$\mathcal {H}_{a,b}(f)(z)=\frac {\varGamma (a+1)}{\varGamma (b+1)}\int _{0}^{1}\frac {f(t)(1-t)^{b}}{(1-tz)^{a+1}} \,dt,$$ where $a$ and $b$ are non-negative real numbers. In particular, for $a=b=\beta ,\nobreakspace {}\mathcal {H}_{a,b}$ becomes the generalized Hilbert operator $\mathcal {H}_\beta $, and $\beta =0$ gives the classical Hilbert operator $\mathcal {H}$. In this article, we find conditions on $a$ and $b$ such that $\mathcal {H}_{a,b}$ is bounded on Dirichlet-type spaces $S^{p}$, $0 \lt p \lt 2$, and on Bergman spaces $A^{p}$, $2 \lt p \lt \infty .$ Also we find an upper bound for the norm of the operator $\mathcal {H}_{a,b}$. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).