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This thesis studies several problems dealing with weighted inequalities and vector-valued operators. A weight is a nonnegative locally integrable function, and weighted inequalities refers to studying a given operator's continuity from one weighted Lebesgue space to another. The case where the underlying measure of both Lebesgue spaces is given by the same weight is known as a one weight inequality and the case where the weights are different is called a two weight inequality. These types of inequalities appear naturally in harmonic analysis from attempts to extend classical results to function spaces where the underlying measure is not necessarily Lebesgue measure. For most operators from harmonic analysis, Muckenhoupt weights represent the class of weights for which a one weight inequality holds. Chapters II and III study questions involving these weights. In particular, Chapter II focuses on determining the sharp dependence of a vector-valued singular integral operator's norm on a Muckenhoupt weight's characteristic; we determine that the vector-valued operator recovers the scalar dependence. Chapter III presents material from a joint work with M. Lacey. Specifically, in this chapter we estimate the weak-type norms of a simple class of vector-valued operators, but are unable to obtain a sharp result. The final two chapters consider two weight inequalities. Chapter IV characterizes the two weight inequality for a subset of the vector-valued operators considered in Chapter III. The final chapter presents examples to argue there is no relationship between the Hilbert transform and the Hardy-Littlewood maximal operator in the two weight setting; the material is taken from a joint work with M. Reguera.

Finite frames are special collections of vectors utilized in Harmonic Analysis and Digital Signal Processing. In this thesis, geometric aspects and construction techniques are considered for the family of k-vector frames in Fn = Rn or Cn sharing a fixed frame operator (denoted Fk(E, Fn), where E is the Hermitian positive definite frame operator), and also the subfamily of this family obtained by fixing a list of vector lengths (denoted Fk µ(E, Fn), where µ is the list of lengths). The family Fk(E, Fn) is shown to be diffeomorphic to the Stiefel manifold Vn(Fk), and Fk µ(E, Fn) is shown to be a smooth manifold if the list of vector lengths µ satisfy certain conditions. Calculations for the dimensions of these manifolds are also performed. Finally, a new construction technique is detailed for frames in Fk(E, Fn) and Fk µ(E, Fn).

Inaug.-diss.--Göttingen. / Vita.

A study of exceptional sets in a finite product of Brelot spaces is made. The principal results obtained are a convergence theorem for decreasing sequences of n-superharmonic functions and an extension theorem for positive n-superharmonic functions. Similar results are obtained for plurisuperharmonic functions.

In this work we will study Hardy-Littlewood maximal function and maximal operator, basing on both classical and most up to date works. In the first chapter we will give definitions for different types of those objects and consider some of their most important properties. The second chapter is entirely devoted to an overview of the fundamental properties of Hardy-Littlewood maximal function, which are strong (p, p) and weak (1, 1) inequalities. Here we list the most actual results on this inequalities in correspondence to the way the maximal func-tion is defined. The third chapter presents the theorem on asymptotic behavior of the lower bound of the constant in the weak-type (1, 1) inequality for the maximal function associated with cubes of Rd, then the dimension d tends to infinity. In the last chapter a method forcomputing constant c, appearing in the main theorem of chapter 3, is given. / <p>QC 20140527</p>

We say that a restriction theorem holds for a curve (gamma) (t) in (//R)('n) if for all f(epsilon) ((//R)('n)) and for some p and q, there is a constant C(,p,q) such that / (VBAR)(VBAR) f (VBAR) (,(gamma)) (VBAR)(VBAR) (,L('q)(du)) (LESSTHEQ) C(,p,q) (VBAR)(VBAR) f (VBAR)(VBAR) (,L('P)((//R)('n))). / In Chapter 1, we prove restriction theorems for non-compact plane curves with non-negative affine curvature when 1 (LESSTHEQ) p < 4/3 and / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / There is an analogous result for space curves in the same chapter. / The Hilbert transform along the curve (gamma) is defined by / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / In Chapter 2, it is shown that when (gamma) has the rapidly decreasing positive affine curvature, H(,(gamma)) is a L('P)-bounded operator for / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 41-44). Also available in electronic format on the Internet.