Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 4 (2015), 765-806.

Inequality for Burkholder's martingale transform

Paata Ivanisvili

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We find the sharp constant C = C(τ,p, EG, EF) of the inequality (G2 + τ2F2)12p CFp, where G is the transform of a martingale F under a predictable sequence ε with absolute value 1, 1 < p < 2, and τ is any real number.

Article information

Anal. PDE, Volume 8, Number 4 (2015), 765-806.

Received: 8 March 2014
Revised: 1 February 2015
Accepted: 25 March 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis 47A30: Norms (inequalities, more than one norm, etc.)

martingale transform martingale inequalities Monge–Ampère equation torsion least concave function concave envelopes Bellman function developable surface


Ivanisvili, Paata. Inequality for Burkholder's martingale transform. Anal. PDE 8 (2015), no. 4, 765--806. doi:10.2140/apde.2015.8.765.

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  • R. Bañuelos and P. Janakiraman, “$L\sp p$-bounds for the Beurling–Ahlfors transform”, Trans. Amer. Math. Soc. 360:7 (2008), 3603–3612.
  • R. Bañuelos and P. J. Méndez-Hernández, “Space-time Brownian motion and the Beurling–Ahlfors transform”, Indiana Univ. Math. J. 52:4 (2003), 981–990.
  • R. Bañuelos and A. Osękowski, “Burkholder inequalities for submartingales, Bessel processes and conformal martingales”, Amer. J. Math. 135:6 (2013), 1675–1698.
  • R. Bañuelos and G. Wang, “Sharp inequalities for martingales with applications to the Beurling–Ahlfors and Riesz transforms”, Duke Math. J. 80:3 (1995), 575–600.
  • N. Boros, P. Janakiraman, and A. Volberg, “Perturbation of Burkholder's martingale transform and Monge–Ampère equation”, Adv. Math. 230:4-6 (2012), 2198–2234.
  • D. L. Burkholder, “Boundary value problems and sharp inequalities for martingale transforms”, Ann. Probab. 12:3 (1984), 647–702.
  • K. P. Choi, “A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L\sp p(0,1)$”, Trans. Amer. Math. Soc. 330:2 (1992), 509–529.
  • P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, and P. B. Zatitskiy, “Bellman function for extremal problems in BMO”, 2012. To appear in Trans. AMS.
  • P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, and P. B. Zatitskiy, “On Bellman function for extremal problems in BMO”, C. R. Math. Acad. Sci. Paris 350:11-12 (2012), 561–564.
  • P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, and P. B. Zatitskiy, “Bellman function for extremal problems on BMO, II: Evolution”. In preparation.
  • A. Reznikov, V. Vasyunin, and V. Volberg, “Extremizers and Bellman function for martingale weak type inequality”, preprint, 2013.
  • L. Slavin and V. Vasyunin, “Sharp results in the integral-form John–Nirenberg inequality”, Trans. Amer. Math. Soc. 363:8 (2011), 4135–4169.
  • V. Vasyunin and A. Volberg, “Burkholder's function via Monge–Ampère equation”, Illinois J. Math. 54:4 (2010), 1393–1428.