WEIGHTED NORM INEQUALITIES FOR MAXIMAL OPERATORS
Abstract
The thesis consists of four chapters. In the first chapter, we develop a necessary and sufficient condition on w(x) for the Hardy-Littlewood maximal operator to be bounded from L('p,q) (Lorentz space with weight w(x)) to L('p,(INFIN)). Our theorem generalizes the weak type result of B. Muckenhoupt on weighted L('p) spaces. In chapter two we consider the L(p,q) generalization of the Hardy-Littlewood maximal operator, M(,p,q), which was first introduced by E. M. Stein. We show that M(,p,q) is bounded from L('p,q) to L('p,(INFIN)) if and only if w (ELEM) A(,1) (the Muckenhoupt A(,1) condition) when 1 (LESSTHEQ) q (LESSTHEQ) p, and w = 0 almost everywhere when 1 < p < q (LESSTHEQ) (INFIN). In the third chapter we study a dyadic analogue T(,n) of an operator associated with the almost everywhere convergence of Fourier series. We prove that, for p > 1, the maximal operator M* defined by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is of strong type (p,p). The techniques of L. Carleson and R. Hunt are used. We prove in the last chapter that, for p > 1, if the non-negative locally integrable function w(x) satisfies the Muckenhoupt's A(,p) condition, then the maximal operator M* as defined in chapter three is bounded from L(,w) (L('p) space with weight w(x) ) to L(,w). Our proof follows the lines of proof in trigonometric case.
Degree
Ph.D.
Subject Area
Mathematics
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