Weighted inequalities via dyadic operators and a learning theory approach to compressive sensing

The first part of this dissertation explores the application of dominating operators in harmonic analysis by sparse operators. We present preliminary results on dominating certain operators by sparse and analogous operators, some known and some new. These domination results lead to weighted inequalities for Calderon-Zygmund operators and the Hardy-Littlewood maximal operator, fractional integral operators (Riesz potentials) and the fractional maximal operator, commutators of fractional integral operators with multiplication operators, oscillatory integral operators and random discrete Hilbert transforms. The oscillatory integrals are built by polynomial modulation of Calderon-Zygmund kernels. For random discrete Hilbert transforms, these are the first results of their kind. In the second part, we explore the utility of learning theory in one-bit sensing. We effectively estimate the VC-dimension of hemispheres relative to sparse vectors, which allows us to employ learning theory techniques to control an empirical process. This control yields the (1-bit) Restricted Isometry Property with high probability. With these methods, we analyze the effects of certain noise models on the acquisition scheme.Ph.D