Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

Journal article (2018)

Authors

I.S. Yaroslavtsev Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2018 I.S. Yaroslavtsev
DOI: https://doi.org/10.4064/sm170329-25-8
Fourier multipliers UMD Banach spaces Hilbert transform Lévy process Burkholder function Differential subordination Purely discontinuous martingales Stochastic integration Weak differential subordination Sharp estimates
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Published Date

2018

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

We introduce the notion of weak differential subordination for martingales, and show that a Banach space X is UMD if and only if for all p ∈ (1, ∞) and all purely discontinuous X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate E || N∞ ||p ≤ CpE|| M∞ ||p. As a corollary we derive a sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.

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