Riesz transforms, function spaces, and weighted estimates for Schrödinger operators with non-negative potentials

The main aim of this thesis is to obtain estimates for Riesz transforms associated to the Schrödinger operator with non-negative potentials on various function spaces over the Euclidean spaces. Our results concern the first- and second-order Riesz transforms on the weighted Lp spaces, the Hardy spaces, and the Morrey spaces. We describe our main results briefly in the following. -- We show that the Lp boundedness of the first-order Riesz transforms within a range of exponents is equivalent, firstly to their boundedness on the weighted Lp spaces within a specific range of both p and (Muckenhoupt) weights, and secondly to their boundedness on the Morrey spaces Lp,λ within a specific range of parameters p and λ. -- On specialising to the case where the potential satisfies a reverse Hölder inequality up to some exponent, we show that the second-order Riesz transforms are bounded on three classes of function spaces: the weighted Lebesgue spaces, the Hardy spaces associated to the Schrödinger operator, and the Morrey spaces, again within a specific range of parameters; this time depending on the reverse Hölder exponent. This is achieved through some new estimates on the heat kernel (associated to the Schrödinger operator) that we derive within this context. These estimates involve the time and the spatial derivatives and have extra global decay over the usual Gaussian. -- In this setting we also study a class of weights that generalise the Muckenhoupt weights and are adapted to the Schrödinger operator in a certain sense. We develop some new good-λ estimates that provide a systematic framework for investigating operators lacking kernel regularity on Lp spaces with these weights.