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arXiv · 1108.3109

Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity (m,n)

Jean Carlo Moraes·María Cristina 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 positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) 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. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). 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_2-characteristic of w^{2t} and polynomially in the complexity (m,n), recovering a result of Beznosova for the (0,0)-complexity case.

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Jean Carlo Moraes, María Cristina Pereyra. 2013-06-27. Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity (m,n). https://doi.org/10.5565/publmat_57213_01

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