Weighted estimates for dyadic operators with complexity

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 L²(w)-norm of a paraproduct with complexity $(m,n)$ and the dual paraproduct associated to a function $b\\in BMO$, depends linearly on the A₂-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and polynomially in the complexity. Moreover we prove that the L²(w)-norm of the composition of these operators depends linearly on the A₂-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 L²(w) hold. Also we prove that the L²-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 \π_b is bounded from L²(u) into L²(v) if and only if the weights satisfies the joint A₂ condition.

Name thesis:degree_name

Mathematics

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics & Statistics

Year

2012

Author dc:creator

• Moraes, Jean Carlo Pech de

Contributors dc:contributor

• Pereyra, Maria Cristina
• Maria Cristina Pereyra
• Matthew D. Blair
• Jens Lorenz
• Carlos Pérez

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

Language dc:language

English

Identifier

https://digitalrepository.unm.edu/math_etds/33

OAI identifier oai:identifier

oai:digitalrepository.unm.edu:math_etds-1032