Multiplier theorems, square function estimates, and Bochner-Riesz means associated with rough domains

This thesis contains results of the author from [12], [13], [14], and [15]. In the first part of the thesis, we will prove a characterization of restricted strong type (p; p) boundedness of multiplier operators whose multiplier is a radial function on R3 supported compactly away from the origin, in the range 1 < p < 13/12. This result complements a result of Heo, Nazarov, and Seeger, who obtained a characterization of radial Fourier multiplier operators bounded on Lp(Rd) in dimensions d > 4 for the range 1 < p < (2d--2)/(d+1).;In the second part of the thesis, we introduce and define Bochner-Riesz multipliers associated with convex planar domains. Such multipliers were first studied by Seeger and Ziesler, and we discuss their results as background. We then discuss new results addressing the question of sharpness of Seeger and Ziesler's theorem. We introduce the additive combinatorial notion of "additive energy" of the boundary of a convex domain which we will show gives a sufficient criteria for obtaining improved Lp bounds for Bochner-Riesz multipliers.;In the third part of the thesis, we will introduce general Fourier multipliers associated with convex planar domains and prove a criterion for Lp boundedness of the corresponding multiplier operators. The methods used to obtain multiplier theorems in this section will involve analysis of "half-wave" operators associated with convex domains.;In the fourth part of the thesis, we will discuss a related square function result and obtain new multiplier theorems as a corollary, which we will interpolate with our results from the third part of the thesis to obtain our most general quasiradial multiplier theorem.