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Sparse Domination of Singular Integral Operators

Prof. Parasar Mohanty IIT, Kanpur
Prof. Parasar Mohanty IIT, Kanpur
When Nov 15, 2019
from 04:00 PM to 05:00 PM
Where LH 006
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Abstract: In this series of talks we will discuss the recently developed sparse domination techniques. Essentially this is a technique which allows us to dominate various operators by average of appropriate dyadic operators for which weighted boundedness is easy to handle. The advantage of this technique is that it also provides the sharp bounds. 

Let \(\mathcal{S}\) be a family of cubes. We say that \(\mathcal S\) is \(\eta\)-sparse, \(0<\eta<1\), if for ever cube \(Q\in\mathcal S\) there exists \(E_Q\subset Q\) such that

 \(|E_Q|\ge \eta|Q|,\;\forall Q\)

 \(E_Q\cap E_{Q'}=\emptyset\) if \(Q\neq Q'\).

Denote the sparse operator \(\mathcal{A}_{\mathcal S}f(x)=\sum\limits_Qf_Q\chi_Q(x),\) where \(f_Q=\frac{1}{|Q|}\int_Q f\).

In the first talk we will see the A. Lerner's proof of famous \(A_2\)-conjecture by showing that \(Tf(x)\lesssim \mathcal{A}_{\mathcal S}f(x)\) for some suitably chosen sparse family \(\mathcal S\) and \(T\) being the Calder\('{o}n\)-Zygmund operator. In the subsequent talks by using this techniques we will see weighted bounds for rough singular integrals and multilinear Calder\('{o}\)n-Zygmund operators for non-doubling measures.

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