Why the Riesz transforms are averages of the dyadic shifts

By S. Petermichl, Sergei Treil, Alexander Volberg. 2002

53 citations recorded by the source corpus.

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 Rn —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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2b3a1d2b1b7f85a288905bf479eb8616f2db2c9d

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In Applied Mathematics

Citations in the corpus

A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure (2008, background): “erators of [9]. Citing the specic result here would carry us too far aeld, but this class includes the important Haar shift examples, such as the one found by S. Petermichl [14], and generalized in [15]. Consequently, characterizations are given in [24] and [10] for the Hilbert transform and Riesz t”
The sharp weighted bound for the Riesz transforms (2007, methodology): “This technique then was extended to the Riesz transforms in [13], using the product Haar system over cubes in R.”