| Jones and Makarov [JM95] gave sharp density estimates for harmonic measure using a modified version of Marcinkiewicz integrals called I 0. It was also used by Jones and Smirnov to substantially advance in the Sobolev and quasiconformal removability problems. We generalize I0 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. The version of I 0 that we consider is slightly different than theirs, but is easier to manipulate and has the same applications as theirs.;Our proof is more classical than theirs, decomposes the operator into bite-sized chunks, and allows to "read off" immediately the contribution of each Whitney cube. It is more flexible than the previous one and hence, it should have applications to the aforementioned Sobolev and quasiconformal removability problems, since the geometry and combinatorics of these problems and the estimates proved in this thesis are very similar. The techniques used mainly come from harmonic analysis with a certain combinatorics and probability flavor (e.g. stopping times). |
