Capacity, Carleson Measures, Boundary Convergence, and Exceptional Sets

There is a fundamental relation between the capacity of a set and energy integrals of probability measures supported on that set. If the capacity is small the energy integral will be large; in particular sets of capacity zero cannot support probability measures of …nite energy. Here we develop similar ideas relating capacity to Carleson measures. We show that if a set has small capacity then any probability measure supported on it must have large Carleson embedding constant. In particular sets of capacity zero are exactly the simultaneous null set for all nontrivial Carleson measures. Functions having limited smoothness often exhibit attractive or convenient behavior at most points, the exceptional set being of capacity zero; that is, the good behavior holds quasi-everywhere, henceforth q:e:. Using the relationship between capacity and Carleson measures such a conclusion can be reformulated by saying the function exhibits the good behavior a:e: for every Carleson measure : This can be useful because sometimes it is relatively easy, even tautological, to establish that a property holds a:e:: We will use this viewpoint to give a new approach to results related to boundary behavior of holomorphic and harmonic functions. The dyadic Dirichlet space is a Hilbert space of functions on a dyadic tree. In many ways that space models the classical Dirichlet space, the space of holomorphic functions, f; on the unit disk D for which R D jf 0j < 1: In Sections 3 and 4 we present background material on the dyadic Dirichlet space and the associated theory of Carleson measures. In Section 5 we present our new results on Carleson measures for the dyadic Dirichlet space. Those include the use of Carleson measures to measure capacity and a direct proof of the equivalence of the measure theoretic and capacity theoretic criteria for a Carleson embedding. In Section 6 we show how those results can be used to study boundary behavior and exceptional sets. Roughly, the idea is to work with the tree geometry to construct a majorant of the variation of the function being studied. If this majorant is in a discrete function space, X; then the boundary set on which the majorant is in…nite must be a null set for every X Carleson measure. We then appeal to results of Section 4 to recast this as a statement about the capacity of the exceptional set. The discrete case is a model case in which the proofs are relatively straightforward and the geometric issues we wish to highlight are particularly clear; and in this paper we will focus almost exclusively on that model case. However the