L^P Estimates For A Singular Integral Operator Motivated By Calderxc3Xb3N'S Second Commutator

Abstract

My research centers on Lp estimates for singular integral operators using techniques from real harmonic analysis. In particular I use time-frequency analysis and oscillatory integral theory. Singular integral operators are frequently motivated by, and have potential applications to, non-linear partial differential equations. In my thesis I show a wide range of Lp estimates for an operator motivated by dropping one average in Calderón's second commutator. For comparison by dropping one average in Calderón's first commutator one faces the bilinear Hilbert transform. Lacey and Thiele showed Lp estimates for that operator [11, 12]. By dropping two averages in Calderón's second commutator one obtains the trilinear Hilbert transform. No Lp estimates are known for that operator. The novelty in this thesis is that in order to avoid difficulty of the level of the trilinear Hilbert transform, I choose to view the symbol of the operator as a non-standard symbol.