Burkholder’s function and a weighted 𝐿² bound for stochastic integrals

Bañuelos, Rodrigo; Brzozowski, Michał; Osȩkowski, Adam

doi:10.1090/proc/15136

Burkholder’s function and a weighted $L^2$ bound for stochastic integrals
HTML articles powered by AMS MathViewer

by Rodrigo Bañuelos, Michał Brzozowski and Adam Osȩkowski PDF

Proc. Amer. Math. Soc. 148 (2020), 5013-5028 Request permission

Abstract:

Let $X$ be a continuous-path martingale and let $Y$ be a stochastic integral, with respect to $X$, of some predictable process with values in $[-1,1]$. We provide an explicit formula for Burkholder’s function associated with the weighted $L^2$ bound \begin{equation*} \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}. \end{equation*}

References

Kari Astala, Tadeusz Iwaniec, István Prause, and Eero Saksman, Burkholder integrals, Morrey’s problem and quasiconformal mappings, J. Amer. Math. Soc. 25 (2012), no. 2, 507–531. MR 2869025, DOI 10.1090/S0894-0347-2011-00718-2
Kari Astala, Tadeusz Iwaniec, István Prause, and Eero Saksman, A hunt for sharp $\mathcal L^p$-estimates and rank-one convex variational integrals, Filomat 29 (2015), no. 2, 245–261. MR 3359421, DOI 10.2298/FIL1502245A
Albert Baernstein II and Stephen J. Montgomery-Smith, Some conjectures about integral means of $\partial f$ and $\overline \partial f$, Complex analysis and differential equations (Uppsala, 1997) Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 64, Uppsala Univ., Uppsala, 1999, pp. 92–109. MR 1758918
Rodrigo Bañuelos and Krzysztof Bogdan, Lévy processes and Fourier multipliers, J. Funct. Anal. 250 (2007), no. 1, 197–213. MR 2345912, DOI 10.1016/j.jfa.2007.05.013
Rodrigo Bañuelos and Adam Osȩkowski, On Astala’s theorem for martingales and Fourier multipliers, Adv. Math. 283 (2015), 275–302. MR 3383804, DOI 10.1016/j.aim.2015.07.006
Rodrigo Bañuelos and Adam Osȩkowski, Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space, J. Funct. Anal. 269 (2015), no. 6, 1652–1713. MR 3373431, DOI 10.1016/j.jfa.2015.06.015
Rodrigo Bañuelos and Adam Osękowski, Weighted square function estimates, to appear in Bull. Sci. Math., arXiv:1711.08754v1 [math.PR] 23 Nov 2017.
Richard Bellman, Dynamic programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2010. Reprint of the 1957 edition; With a new introduction by Stuart Dreyfus. MR 2641641
Stephen M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. MR 1124164, DOI 10.1090/S0002-9947-1993-1124164-0
D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
D. L. Burkholder, A sharp and strict $L^p$-inequality for stochastic integrals, Ann. Probab. 15 (1987), no. 1, 268–273. MR 877602
Donald L. Burkholder, A proof of Pełczynśki’s conjecture for the Haar system, Studia Math. 91 (1988), no. 1, 79–83. MR 957287, DOI 10.4064/sm-91-1-79-83
Donald L. Burkholder, Explorations in martingale theory and its applications, École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1–66. MR 1108183, DOI 10.1007/BFb0085167
Claude Dellacherie and Paul-André Meyer, Probabilities and potential. B, North-Holland Mathematics Studies, vol. 72, North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales; Translated from the French by J. P. Wilson. MR 745449
Komla Domelevo and Stefanie Petermichl, Differential subordination under change of law, Ann. Probab. 47 (2019), no. 2, 896–925. MR 3916937, DOI 10.1214/18-AOP1274
Javier Duoandikoetxea, Extrapolation of weights revisited: new proofs and sharp bounds, J. Funct. Anal. 260 (2011), no. 6, 1886–1901. MR 2754896, DOI 10.1016/j.jfa.2010.12.015
T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), no. 6, 1–16 (English, with German and Russian summaries). MR 719167, DOI 10.4171/ZAA/37
Tadeusz Iwaniec, Nonlinear Cauchy-Riemann operators in ${\Bbb R}^n$, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1961–1995. MR 1881026, DOI 10.1090/S0002-9947-02-02914-8
M. Izumisawa and N. Kazamaki, Weighted norm inequalities for martingales, Tohoku Math. J. (2) 29 (1977), no. 1, 115–124. MR 436313, DOI 10.2748/tmj/1178240700
Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473–1506. MR 2912709, DOI 10.4007/annals.2012.175.3.9
Michael T. Lacey, Kabe Moen, Carlos Pérez, and Rodolfo H. Torres, Sharp weighted bounds for fractional integral operators, J. Funct. Anal. 259 (2010), no. 5, 1073–1097. MR 2652182, DOI 10.1016/j.jfa.2010.02.004
Andrei K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912–3926. MR 2770437, DOI 10.1016/j.aim.2010.11.009
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
F. L. Nazarov and S. R. Treĭl′, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), no. 5, 32–162 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 721–824. MR 1428988
F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI 10.1090/S0894-0347-99-00310-0
A. Osękowski, Sharp martingale and semimartingale inequalities, Monografie Matematyczne 72, Birkhäuser, 2012.
Goran Peskir, A change-of-variable formula with local time on surfaces, Séminaire de Probabilités XL, Lecture Notes in Math., vol. 1899, Springer, Berlin, 2007, pp. 69–96. MR 2408999
Stefanie Petermichl and Alexander Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. MR 1894362, DOI 10.1215/S0012-9074-02-11223-X
Armen Vagharshakyan, Recovering singular integrals from Haar shifts, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4303–4309. MR 2680056, DOI 10.1090/S0002-9939-2010-10426-4
Gang Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), no. 2, 522–551. MR 1334160
Janine Wittwer, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000), no. 1, 1–12. MR 1748283, DOI 10.4310/MRL.2000.v7.n1.a1