## Mathematics & Statistics ETDs

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.

Mathematics

Doctoral

#### Department Name

Mathematics & Statistics

Pereyra, Maria Cristina

Blair, Matthew

Lorenz, Jens

#### Third Committee Member

Perez, Carlos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and Fulbright.

English

#### Keywords

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

Dissertation

COinS