Why the Riesz transforms are averages of the dyadic shifts?

Abstract

The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators -so-called dyadic shifts. We show here that the same is true in any $\mathbb{R}^n$ -the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the previous approach, rather than presenting the new one