On zero sets in the Dirichlet space

We study the zeros sets of functions in the Dirichlet space. Using Carleson formula for Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of E \subset \TT with logarithmic capacity zero is the accumulation points of the zeros of a function in the Dirichlet space. The zeros satisfy a growth restriction which depends on $E$.Comment: Journal of Geometric Analysis (2011

Blaschke products and singular functions with prescribed boundary values

AbstractThis paper shows that there exists a Blaschke product having a prescribed radial limit at each point of a prescribed finite subset of the unit circle. In addition, an analogue for singular inner functions is proved; and an extension dealing with tangential limits is established