| Chapter 1. In this section we study the asymptotic properties of solutions to the equation f′′x +fx 2fx+Ef x-Vxf x=0, E>0, treating the potential V(x) as a perturbation. The linear part of this equation has been analyzed quite extensively, yielding results about the spectral theory of one-dimensional Schrödinger operators. Our main result is the following statement: If |V(x)| ≤ C|x|−β, β > , then, for every energy E > 0, almost every perturbed solution asymptotically approaches a solution to the unperturbed equation.; This differs from comparable results in the linear setting in two respects. First, the lower limit on β is as opposed to ½ for the linear Schrödinger operator. Second, the asymptotic description holds for almost every solution at every energy rather than holding for every solution at almost every energy.; Both discrepancies arise from the same basic consequence of nonlinearity: even at a fixed energy E, solutions to the unperturbed nonlinear equation may oscillate with different periods. Their peaks and troughs do not remain in tandem as x → ∞ but instead drift in and out of phase with one another. The stronger decay condition on V is necessary to control the rate at which perturbed solutions “drift” in relation to any given unperturbed solution.; The structure of the almost-everywhere conclusion is derived from a Fourier analysis argument in which the period of unperturbed solutions plays a crucial role. In the linear case, all solutions to the unperturbed equation with a fixed energy have the same period, therefore the value of E is the determining factor. In the nonlinear case, where the period of unperturbed solutions depends on additional parameters, the set of exceptional solutions does not possess such a transparent structure.; Chapter 2. The Muckenhoupt Ap condition concisely describes the class of measures dμ = w(x)dx for which several noteworthy operators (the Hilbert Transform and Hardy-Littlewood maximal function, to name two) are bounded on Lp(dμ). In the setting of vector-valued functions, weights w( x) are defined to take values in the space of nonnegative self-adjoint matrices, and it is natural to ask what condition on matrix weights will continue to guarantee the Lp-boundedness of these operators.; The theory of matrix Ap weights, as developed by Nazarov, Treil, and Volberg, answers this question for singular integral operators. Their approach is quite different from the classical scalar method, eschewing the Hardy-Littlewood maximal function and distributional estimates in favor of Carleson measures and a dyadic embedding operator. Some of the results thus obtained were novel even when restricted to scalars as a special case. In the current work we attempt to perform the opposite feat: employing techniques commonly associated with the scalar case to gain new perspectives on matrix Ap |