Generalized Ces$\grave{a}$ro Operator on Dirichlet Spaces

Abstract:

      In this paper, we study the boundedness of the generalized Ces$\grave{a}$ro operator on the weighted Dirichlet spaces $${\cal D}_{\alpha}=\left\{f\in H(D); ||f||_{{\cal D}_{\alpha}}^{2}=|f(0)|^{2}+\int_{D}|f'(z)|^{2}(1-|z|)^{\alpha}\d m(z)<+\infty \right\},$$ where $-1<\alpha<+\infty$ and $H(D)$ is the class of all holomorphic functions on the unit disc $D$.