A sharp Lp estimate for the total variation process

Adam Osękowski

Statistics & Probability Letters, 2024, vol. 206, issue C

Abstract: For a given p≥2, let X be an Lp bounded martingale and let Y be a martingale of bounded mean oscillation. The paper contains the proof of the estimate ‖∫0∞|d〈X,Y〉t|‖p≤p‖X‖p‖Y‖bmo. The inequality is sharp for each p and the range p≥2 cannot be expanded without additional assumptions on X and Y. The proof rests on the existence of a certain special function, enjoying appropriate size and concavity requirements.

Keywords: Skew bracket; Bounded mean oscillation; Martingale; Best constant (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spl.2023.109997

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