Thesis Defense: Weighted Estimates for One-Sided Calderón-Zygmund Operators and Discrete Harmonic Analogs

This dissertation addresses two questions in harmonic analysis. First, we present a proof of the one-sided A₂ theorem in dimension one with a logarithmic loss. This theorem concerns one-sided Calderón-Zygmund operators whose kernels vanish whenever the input variable lies to the left of the output variable. Such operators are bounded on L²(w) whenever the weight w belongs to the one-sided class of weights. The norm estimate is reduced to testing on indicator functions through a two-weight testing theorem, and the bound follows from a weak-type (1,1) inequality and an extrapolation argument. A localized version on fixed intervals is obtained by introducing adapted weight classes. Second, we construct harmonic extensions to the upper half-integer lattice ℤ × ℕ for a given boundary sequence; these act as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by discrete harmonicity with respect to a two-dimensional Laplacian, a discrete Cauchy-Riemann system, and boundary values involving a new discrete Hilbert transform. The transform is compared to the Riesz-Titchmarsh transform, and Lebesgue bounds are established. An extension to higher-dimensional lattices is also provided. Together, the two parts offer continuous one-sided weighted estimates and a fully discrete harmonic-analytic theory.

Faculty Advisor: Brett Wick