## Weighted estimates for dyadic operators with complexity

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

Title

Weighted estimates for dyadic operators with complexity

Author(s)

Moraes, Jean Carlo Pech de

Advisor(s)

Pereyra, Maria Cristina

Committee Member(s)

Blair, Matthew

Lorenz, Jens

Perez, Carlos

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.

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.

Date

December 2011

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- Mathematics [28]