Ignacio Uriarte-Tuero (Universidad Autonoma de Madrid)
On Marcinkiewicz integrals and harmonic measure
Abstract.
Jones and Makarov gave sharp density estimates for harmonic
measure using
a modified version of Marcinkiewicz integrals called $\tilde
I_{0}$. It
was also used by Jones and Smirnov to substantially advance in
the Sobolev
and quasiconformal removability problems. We generalize $\tilde
I_{0}$ to
make it account for different densities of sets over which to
integrate,
in particular giving a different proof than Jones' and Makarov's
of its
key properties. This should have applications to the
aforementioned
Sobolev and quasiconformal removability problems.
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