Discrete Hilbert transform à la Gundy–Varopoulos

Arcozzi, N.; Domelevo, K.; Petermichl, S.

doi:10.1090/proc/14492

Discrete Hilbert transform à la Gundy–Varopoulos
HTML articles powered by AMS MathViewer

by N. Arcozzi, K. Domelevo and S. Petermichl PDF

Proc. Amer. Math. Soc. 148 (2020), 2433-2446 Request permission

Abstract:

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy–Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.

References

Nicola Arcozzi, Riesz transforms on spheres and compact Lie groups, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–Washington University in St. Louis. MR 2693475
Nicola Arcozzi, Komla Domelevo, and Stefanie Petermichl, Second order Riesz transforms on multiply-connected Lie groups and processes with jumps, Potential Anal. 45 (2016), no. 4, 777–794. MR 3558360, DOI 10.1007/s11118-016-9566-x
Rodrigo Bañuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications, Illinois J. Math. 54 (2010), no. 3, 789–868 (2012). MR 2928339
Rodrigo Bañuelos and Fabrice Baudoin, Martingale transforms and their projection operators on manifolds, Potential Anal. 38 (2013), no. 4, 1071–1089. MR 3042695, DOI 10.1007/s11118-012-9307-8
Rodrigo Bañuelos and Mateusz Kwaśnicki, On the $\ell ^p$-norm of the discrete Hilbert transform, Duke Math. J. 168 (2019), no. 3, 471–504. MR 3909902, DOI 10.1215/00127094-2018-0047
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 Gang Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575–600. MR 1370109, DOI 10.1215/S0012-7094-95-08020-X
Rodrigo Bañuelos and Gang Wang, Orthogonal martingales under differential subordination and applications to Riesz transforms, Illinois J. Math. 40 (1996), no. 4, 678–691. MR 1415025
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
Donald L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), no. 2, 995–1025. MR 1288140
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
Óscar Ciaurri, T. Alastair Gillespie, Luz Roncal, José L. Torrea, and Juan Luis Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017), 109–131. MR 3666807, DOI 10.1007/s11854-017-0015-6
Laura De Carli and Gohin Shaikh Samad, One-parameter groups of operators and discrete Hilbert transforms, Canad. Math. Bull. 59 (2016), no. 3, 497–507. MR 3563731, DOI 10.4153/CMB-2016-028-7
Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
K. Domelevo, A. Osȩkowski, and S. Petermichl, Various sharp estimates for semi-discrete Riesz transforms of the second order, 50 years with Hardy spaces, Oper. Theory Adv. Appl., vol. 261, Birkhäuser/Springer, Cham, 2018, pp. 229–255. MR 3792098
Komla Domelevo and Stefanie Petermichl, Sharp $L^p$ estimates for discrete second order Riesz transforms, Adv. Math. 262 (2014), 932–952. MR 3228446, DOI 10.1016/j.aim.2014.06.003
R. J. Duffin, Basic properties of discrete analytic functions, Duke Math. J. 23 (1956), 335–363. MR 78441
Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
Matts Essén, A superharmonic proof of the M. Riesz conjugate function theorem, Ark. Mat. 22 (1984), no. 2, 241–249. MR 765412, DOI 10.1007/BF02384381
Jacqueline Ferrand, Fonctions préharmoniques et fonctions préholomorphes, Bull. Sci. Math. (2) 68 (1944), 152–180 (French). MR 13411
I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
Maru Guadie and Eugenia Malinnikova, Stability and regularization for determining sets of discrete Laplacian, Inverse Problems 29 (2013), no. 7, 075018, 17. MR 3080479, DOI 10.1088/0266-5611/29/7/075018
Richard F. Gundy and Nicolas Th. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 1, A13–A16 (French, with English summary). MR 545671
H. A. Heilbronn, On discrete harmonic functions, Proc. Cambridge Philos. Soc. 45 (1949), 194–206. MR 30051, DOI 10.1017/s0305004100024713
Rufus Philip Isaacs, A finite difference function theory, Univ. Nac. Tucumán. Revista A. 2 (1941), 177–201. MR 0006391
David Jerison, Lionel Levine, and Scott Sheffield, Internal DLA and the Gaussian free field, Duke Math. J. 163 (2014), no. 2, 267–308. MR 3161315, DOI 10.1215/00127094-2430259
S. Kak, The discrete Hilbert transform, Proc. IEEE 58 (1970) 585–586.
Enrico Laeng, Remarks on the Hilbert transform and on some families of multiplier operators related to it, Collect. Math. 58 (2007), no. 1, 25–44. MR 2310545
Gabor Lippner and Dan Mangoubi, Harmonic functions on the lattice: absolute monotonicity and propagation of smallness, Duke Math. J. 164 (2015), no. 13, 2577–2595. MR 3405594, DOI 10.1215/00127094-3164790
Françoise Lust-Piquard, Dimension free estimates for discrete Riesz transforms on products of abelian groups, Adv. Math. 185 (2004), no. 2, 289–327. MR 2060471, DOI 10.1016/j.aim.2003.07.002
V. Matsaev and M. Sodin, Distribution of Hilbert transforms of measures, Geom. Funct. Anal. 10 (2000), no. 1, 160–184. MR 1748919, DOI 10.1007/s000390050005
V. I. Macaev, Volterra operators obtained from self-adjoint operators by perturbation, Dokl. Akad. Nauk SSSR 139 (1961), 810–813 (Russian). MR 0136997
Adam Osękowski, Sharp martingale and semimartingale inequalities, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 72, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2964297, DOI 10.1007/978-3-0348-0370-0
S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 165–179. (errata insert). MR 312140, DOI 10.4064/sm-44-2-165-179
Lillian Beatrix Pierce, Discrete analogues in harmonic analysis, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Princeton University. MR 2713096
Nicolas Privault, Stochastic analysis in discrete and continuous settings with normal martingales, Lecture Notes in Mathematics, vol. 1982, Springer-Verlag, Berlin, 2009. MR 2531026, DOI 10.1007/978-3-642-02380-4
Nicolas Privault, Stochastic finance, Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL, 2014. An introduction with market examples. MR 3202743
Philip E. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, vol. 21, Springer-Verlag, Berlin, 2005. Second edition. Version 2.1; Corrected third printing. MR 2273672, DOI 10.1007/978-3-662-10061-5
Elias M. Stein and Stephen Wainger, Discrete analogues in harmonic analysis. I. $l^2$ estimates for singular Radon transforms, Amer. J. Math. 121 (1999), no. 6, 1291–1336. MR 1719802
E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis. II. Fractional integration, J. Anal. Math. 80 (2000), 335–355. MR 1771530, DOI 10.1007/BF02791541
I. È. Verbickiĭ, Estimate of the norm of a function in a Hardy space in terms of the norms of its real and imaginary parts, Mat. Issled. 54 (1980), 16–20, 164–165 (Russian). Linear operators. MR 577230