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

Ivan Yaroslavtsev

doi:10.4064/sm170329-25-8

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

Ivan Yaroslavtsev^*

^*Corresponding author for this work

Analysis

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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 ≤ C_pE|| 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.

Original language	English
Pages (from-to)	269-301
Number of pages	33
Journal	Studia Mathematica
Volume	243
Issue number	3
DOIs	https://doi.org/10.4064/sm170329-25-8
Publication status	Published - 2018

Keywords

Burkholder function
Differential subordination
Fourier multipliers
Hilbert transform
Lévy process
Purely discontinuous martingales
Sharp estimates
Stochastic integration
UMD Banach spaces
Weak differential subordination

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10.4064/sm170329-25-8

1703.07817Accepted author manuscript, 319 KB

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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.",

keywords = "Burkholder function, Differential subordination, Fourier multipliers, Hilbert transform, L{\'e}vy process, Purely discontinuous martingales, Sharp estimates, Stochastic integration, UMD Banach spaces, Weak differential subordination",

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T1 - Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

AU - Yaroslavtsev, Ivan

PY - 2018

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N2 - 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.

AB - 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.

KW - Burkholder function

KW - Differential subordination

KW - Fourier multipliers

KW - Hilbert transform

KW - Lévy process

KW - Purely discontinuous martingales

KW - Sharp estimates

KW - Stochastic integration

KW - UMD Banach spaces

KW - Weak differential subordination

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DO - 10.4064/sm170329-25-8

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SN - 0039-3223

VL - 243

SP - 269

EP - 301

JO - Studia Mathematica

JF - Studia Mathematica

IS - 3

ER -

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

Abstract

Keywords

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Martingales and stochastic calculus in Banach spaces

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