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CAPACITY, CARLESON MEASURES, BOUNDARY CONVERGENCE, AND EXCEPTIONAL SETS
 

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 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.

  

Source: Arcozzi, Nicola - Dipartimento di Matematica, UniversitÓ di Bologna

 

Collections: Mathematics