Riesz Transform estimates in the absence of a preservation condition and applications to the Dirichlet Laplacian

R. Strichartz in [68] asked whether the Lp boundedness of the Riesz transform observed on Rn could be extended to a reasonable class of non-compact manifolds. Many partial answers have been given since. One such answer given by Auscher, Coulhon, Duong and Hofmann in [5] tied the Lp boundedness of the Riesz transform to the Lp boundedness of the Gaffney inequality. Their result was for p > 2 and held on noncompact manifolds satisfying doubling and Poincaré conditions, along with a stochastic completeness or preservation condition. In this thesis the results of [5] are adapted to prove Lp bounds, p > 2, for Riesz transform variations in cases where a preservation condition does not hold. To compensate for the lack of a preservation condition, two new conditions are required. The results are general enough to apply in a large number of circumstances. Two extensions on this result are additionally presented. The first extension is to non-doubling domains. This extension is specifically in the circumstance of a manifold with boundary and Dirichlet boundary conditions. An added benefit of this non-doubling extension is that the Poincaré inequality is no longer required near the boundary. The second extension shows that the weighted Lp boundedness of the Riesz transform observed on Rn can also be extended in some degree to a reasonable class of non-compact manifolds. This second extension includes generalised deriving of weight classes associated to skewed maximal functions and other operators. This thesis also contains applications to the case of the Dirichlet Laplacian on various subsets of Rn. The overall work and particularly the application are motivated by recent results from Killip, Visan and Zhang in [48].