Ces\`aro 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.