Thesis Defense: Wavelet Representation of Singular Integral Operators
Abstract: The idea of representing singular integral operators as averages dyadic shifts has proven fruitful since Petermichl's representation of the Hilbert transform, and its generalization by Hytönen to prove the A_2 conjecture. An alternate approach to wavelet representation was provided by Di Plinio, Wick, and Williams (2022) in which the random dyadic grids are replaced by zero-complexity wavelet projections, providing finer control of smooth operators. The goal of this thesis is to provide two main generalizations of this result: firstly in the ambient setting of spaces of homogeneous type, and secondly in the smoothness of the operators to Dini continuity. The SHT statement is made only in the case of fractional-order smoothness, while the Dini case strengthens even the original statement for power moduli of continuity, by sharpening the loss in smoothness of the adapted wavelets to be precisely double-logarithmic. Both results are new and permit broader applications of the continuous representation theorem. We will conclude by pursuing one such application, a T(1) theorem for compact operators.
Advisor: Brett Wick