TY - JOUR

T1 - The Hilbert Transform and Orthogonal Martingales in Banach Spaces

AU - Osękowski, Adam

AU - Yaroslavtsev, Ivan

PY - 2019

Y1 - 2019

N2 - Let X be a given Banach space, and let M and N be two orthogonal X-valued local martingales such that N is weakly differentially subordinate to M. The paper contains the proof of the estimate E Ψ (Nt) ≤ CΦ, Ψ, X E Φ (Mt), t ≥ 0, where Φ, Ψ: X → R+ are convex continuous functions and the least admissible constant CΦ, Ψ, X coincides with the Φ, Ψ -norm of the periodic Hilbert transform. As a corollary, it is shown that the Φ, Ψ -norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if Φ is symmetric. We also prove that under certain natural assumptions on Φ and Ψ, the condition CΦ, Ψ, X < ∞ yields the UMD property of the space X. As an application, we provide comparison of Lp-norms of the periodic Hilbert transform to Wiener and Paley-Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

AB - Let X be a given Banach space, and let M and N be two orthogonal X-valued local martingales such that N is weakly differentially subordinate to M. The paper contains the proof of the estimate E Ψ (Nt) ≤ CΦ, Ψ, X E Φ (Mt), t ≥ 0, where Φ, Ψ: X → R+ are convex continuous functions and the least admissible constant CΦ, Ψ, X coincides with the Φ, Ψ -norm of the periodic Hilbert transform. As a corollary, it is shown that the Φ, Ψ -norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if Φ is symmetric. We also prove that under certain natural assumptions on Φ and Ψ, the condition CΦ, Ψ, X < ∞ yields the UMD property of the space X. As an application, we provide comparison of Lp-norms of the periodic Hilbert transform to Wiener and Paley-Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

UR - http://www.scopus.com/inward/record.url?scp=85138897690&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz187

DO - 10.1093/imrn/rnz187

M3 - Article

SN - 1073-7928

VL - 2021 (2021)

SP - 11670

EP - 11730

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 15

ER -