Carleson measures and elliptic boundary value problems

Wednesday, May 30, 2012 - 2:00pm - 2:50pm
Keller 3-180
Jill Pipher (Brown University)
L. Carleson introduced the measures which bear his name to solve an interpolation
problem for analytic functions (Ann. of Math.,1962), establishing their relationship with
the existence of nontangential limits at the boundary. These measures were subsequently
understood within the larger context of duality of tent spaces. Carleson measures have
played a fundamental role in the theory of elliptic boundary value problems, especially
in determining solvability of boundary value problems in the context of non-smooth real
or complex coecient operators. The appearance of Carleson measures in this theory
is quite natural: solvability in Lp of an elliptic Dirichlet problem is determined by the
property of the \weight or elliptic measure, and weight classes are closely connected
to the function space BMO and even have Carleson type characterizations. However,
there is an extraordinary variety of ways in which the subtelty of the Carleson measure
characterization emerges in elliptic theory. In these lectures, we describe some classical
and some modern results in this subject which illustrate this theme.
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