Cesàro Summability of Taylor Series in Weighted Dirichlet Spaces

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $$(C,\alpha )$$ -summable to the function in the norm of the space, provided that $$\alpha >1/2$$ . We further show that the constant 1/2 is sharp, in marked contrast with the classical case of the disk algebra.