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Weighted estimates for dyadic paraproducts and $\mathbf{t}$- multipliers with complexity $(m,n)$

Jean Carlo Moraes and María Christina Pereyra

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Abstract

We extend the definitions of dyadic paraproduct and $t$-Haar multipliers to dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$ natural numbers. We use the ideas developed by Nazarov and Volberg to prove that the weighted $L^2(w)$\guio{norm} of a paraproduct with complexity~$(m,n)$, associated to a function $b\in \mathit{BMO}^d$, depends linearly on the $A^d_2$-characteristic of the weight~$w$, linearly on the $\mathit{BMO}^d$-norm of $b$, and polynomially on the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct due to Beznosova. We also prove that the $L^2$-norm of a $t$-Haar multiplier for any $t\in\mathbb{R}$ and weight~$w$ is a multiple of the square root of the $C^d_{2t}$-characteristic of $w$ times the square root of the $A^d_2$-characteristic of $w^{2t}$, and is polynomial in the complexity.

Article information

Source
Publ. Mat. Volume 57, Number 2 (2013), 265-294.

Dates
First available in Project Euclid: 12 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.pm/1386857697

Mathematical Reviews number (MathSciNet)
MR3114770

Zentralblatt MATH identifier
1288.42014

Subjects
Primary: 42C99: None of the above, but in this section
Secondary: 47B38: Operators on function spaces (general)

Keywords
Operator-weighted inequalitiese dyadic paraproduct $A_p$-weights Haar multipliers

Citation

Moraes, Jean Carlo; Pereyra, María Christina. Weighted estimates for dyadic paraproducts and $\mathbf{t}$- multipliers with complexity $(m,n)$. Publ. Mat. 57 (2013), no. 2, 265--294. https://projecteuclid.org/euclid.pm/1386857697.


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