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Weighted estimates for dyadic operators with complexity

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Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/17491

Weighted estimates for dyadic operators with complexity

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Title: Weighted estimates for dyadic operators with complexity
Author: Moraes, Jean Carlo Pech de
Advisor(s): Pereyra, Maria Cristina
Committee Member(s): Blair, Matthew
Lorenz, Jens
Perez, Carlos
Department: University of New Mexico. Dept. of Mathematics and Statistics
Subject(s): dyadic operators, weights, complexity, Muckenhoupt
LC Subject(s): Measure theory.
Linear operators.
Inequalities (Mathematics)
Integrals, Haar.
Lebesgue integral.
Degree Level: Doctoral
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.
Graduation Date: December 2011
URI: http://hdl.handle.net/1928/17491

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