 
Summary: CAPACITY, CARLESON MEASURES, BOUNDARY
CONVERGENCE, AND EXCEPTIONAL SETS
N. ARCOZZI, R. ROCHBERG, AND E. SAWYER
1. Introduction and Summary
There is a fundamental relation between the capacity of a set and energy inte
grals 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 prob
ability 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 par
ticular sets of capacity zero are exactly the simultaneous null set for all nontrivial
Carleson measures.
Functions having limited smoothness often exhibit attractive or convenient be
havior at most points, the exceptional set being of capacity zero; that is, the good
behavior holds quasieverywhere, 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.
