On singular integral and martingale transforms

By Stefan Geiß, Stephen Montgomery-Smith, Eero Saksman. 2007

99 citations recorded by the source corpus.

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on L p X (R 2 ) with p ∈ (1, ∞). Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals p * — 1 with p * := max{p, (p/(p - 1))}, where the novelty is the lower bound.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19f16e558bf479069c3dbe1b7ef01e1bb2089bf8

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In Applied Mathematics

1. Rodrigo Bañuelos
2. Bartłlomiej Dyda
3. Mateusz Kwaśnicki
4. Krzysztof Bogdan
5. Nicolai V. Krylov

In Mathematical Physics

1. Rodrigo Bañuelos
2. Bartłlomiej Dyda
3. Mateusz Kwaśnicki
4. Nicolai V. Krylov

Citations in the corpus

• The foundational inequalities of D.L. Burkholder and some of their ramifications (2010, methodology): “In the [76], Geiss, Montgomery-Smith and Saksman used arguments similar to those used by Bourgain in [27] to show that Burkholder’s UMD property is equivalent to the boundedness of the Hilbert transform to prove that the Lp(IR)-norms of 2RjRk, and R2 j − R2 k, j 6= k, are bounded below by (p∗ − 1).”