Potential theory and boundary behavior in the Drury-Arveson space

We develop a notion of capacity for the Drury-Arveson space $H_d^2$ of holomorphic functions on the Euclidean unit ball. We show that every function in $H_d^2$ has a non-tangential limit (in fact Korányi limit) at every point in the sphere outside of a set of capacity zero. Moreover, we prove that the capacity zero condition is sharp, and that it is equivalent to being totally null for $H_d^2$. We also provide applications to cyclicity. Finally, we discuss generalizations of these results to other function spaces on the ball.