Non-probabilistic proof of the A_2 theorem, and sharp weighted bounds for the q-variation of singular integrals

TL;DR

This paper provides a simple, non-probabilistic proof that Calderon-Zygmund operators can be dominated by positive dyadic operators and extends this to q-variation, leading to sharp weighted bounds.

Contribution

It introduces an elementary proof for domination of Calderon-Zygmund operators and applies it to q-variation, improving understanding of weighted inequalities.

Findings

Elementary proof of operator domination

Extension to q-variation of operators

Sharp weighted inequalities achieved

Abstract

Any Calderon-Zygmund operator T is pointwise dominated by a convergent sum of positive dyadic operators. We give an elementary self-contained proof of this fact, which is simpler than the probabilistic arguments used for all previous results in this direction. Our argument also applies to the q-variation of certain Calderon-Zygmund operators, a stronger nonlinearity than the maximal truncations. As an application, we obtain new sharp weighted inequalities.