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Mathematics > Classical Analysis and ODEs

Title: Sharp inequalities for linear combinations of orthogonal martingales

Abstract: For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of the form $aX+bY$ with $a, b$ real numbers. The best $L^p$ constant is equal to the norm of the operator $aI+bH$ from $L^p$ to $L^p$, where $H$ is the Hilbert transform on the circle or real line. The values of these norms were found by Hollenbeck, Kalton and Verbitsky \cite{HKV}.
Comments: 10 pages
Subjects: Classical Analysis and ODEs (math.CA); Probability (math.PR)
MSC classes: 60G44, 42A45
Cite as: arXiv:1803.04570 [math.CA]
  (or arXiv:1803.04570v1 [math.CA] for this version)

Submission history

From: Kai Zhu [view email]
[v1] Mon, 12 Mar 2018 23:17:00 GMT (9kb)