Publication Date

2-1-2012

Abstract

We extend the definitions of dyadic paraproduct, dual dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ positive integers. We will use the ideas developed by Nazarov and Volberg in \\cite{NV} to prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\in BMO$, depends linearly on the $A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the $L^2(w)$-norm of the composition of these operators depends linearly on the $A_2$-characteristic of the weight $w$, quadratic on the $BMO$-norm of $b$, and polynomially in the complexity. The argument for the paraproduct provides a new proof of the linear bound for the dyadic paraproduct \\cite{Be1} (the one with complexity $(0,0)$). Paraproducts and their adjoints are examples of Haar shift multipliers of type 2 and 3. We adapt the Nazarov and Volberg method to show that for certain Haar shift multipliers of type 4 and complexity $(m,n)$ the same type of bounds in $L^2(w)$ hold. Also we prove that the $L^2$-norm of a $t$-Haar multiplier for any $t$ and weight $w$ depends on the square root of the $C_{2t}$-characteristic of $w$ times the square root of the $A_q$-characteristic of $w^{2t}$ and polynomially in the complexity $(m,n)$, recovering a result of Beznosova \\cite{Be} for the $(0,0)$-complexity case. Last, we prove that for a pair of weights $u$ and $v$ and a class of locally integrable function $b$ that satisfies certain conditions, the dyadic paraproduct $\\pi_b$ is bounded from $L^2(u)$ into $L^2(v)$ if and only if the weights satisfies the joint $A_2$ condition.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Maria Cristina Pereyra

Second Committee Member

Matthew D. Blair

Third Committee Member

Jens Lorenz

Fourth Committee Member

Carlos Pérez

Project Sponsors

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and Fulbright.

Language

English

Keywords

Measure theory, Linear operators, Inequalities (Mathematics), Integrals, Haar, Lebesgue integral.

Document Type

Dissertation

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