Wavelet Representation of Singular Integral Operators

Abstract

The idea of representing singular integral operators as averages dyadic shifts has proven fruitful since Petermichl\u27s representation of the Hilbert transform, and its generalization by Hyt\ onen to prove the $A_2$ conjecture. These results employ a random dyadic decomposition of the operator in terms of Haar shifts of all complexities. 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 and a more efficiently computable representation. The goal of this thesis is to provide two main generalizations of this continuous representation theorem; Both results are new and permit broader applications of the wavelet representation. First, we loosen the smoothness required of the Calder\\u27on--Zygmund operators to be Dini-type. This strengthens even the original statement for H\ older moduli of continuity by improving the loss of smoothness of the adapted wavelets to be precisely double-logarithmic. In general, the double log-Dini condition is likely not sharp, but is essential to the construction of a representation from wavelet averaging. We also consider operators in the ambient setting of spaces of homogeneous type, using wavelets adapted to dyadic grids constructed by Auscher and H\ ytonen. Because of the purely geometric nature of such spaces, this statement is made only in the case of fractional-order smoothness. Consequently, the statement is purely $T(1)$-type, which is still sufficient for proving new applications applications such as representations for compact Calder\\u27on-Zygmund operators