Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces
I.S. Yaroslavtsev (TU Delft - 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 ≤ 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.