A probabilistic approach to Hilbert transforms on free group von Neumann algebras

Gałązka, Tomasz; Osękowski, Adam

doi:10.1090/proc/15732

A probabilistic approach to Hilbert transforms on free group von Neumann algebras
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by Tomasz Gałązka and Adam Osękowski PDF

Proc. Amer. Math. Soc. 150 (2022), 2861-2877 Request permission

Abstract:

The paper contains a probabilistic proof of the $L^p$-boundedness of the Hilbert transform in the context of a free group von Neumann algebra $VN(\mathbb {F}_q)$. The argument rests on noncommutative version of good-$\lambda$ inequalities and yields a tight $L\log L$ order of the $L^p$ norms as $p\to 1^+$ and $p\to \infty$.

References

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
D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226, DOI 10.1214/aop/1176993220
Colin Bennett, A best constant for Zygmund’s conjugate function inequality, Proc. Amer. Math. Soc. 56 (1976), 256–260. MR 402393, DOI 10.1090/S0002-9939-1976-0402393-4
Colin Bennett, Ronald A. DeVore, and Robert Sharpley, Weak-$L^{\infty }$ and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601–611. MR 621018, DOI 10.2307/2006999
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
Burgess Davis, On the weak type $(1,\,1)$ inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307–311. MR 348381, DOI 10.1090/S0002-9939-1974-0348381-6
Leszek Gawarecki and Vidyadhar Mandrekar, Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations, Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2560625, DOI 10.1007/978-3-642-16194-0
Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082
Prabhu Janakiraman, Best weak-type $(p,p)$ constants, $1\leq p\leq 2$, for orthogonal harmonic functions and martingales, Illinois J. Math. 48 (2004), no. 3, 909–921. MR 2114258
Yong Jiao, Adam Osękowski, and Lian Wu, Inequalities for noncommutative differentially subordinate martingales, Adv. Math. 337 (2018), 216–259. MR 3853050, DOI 10.1016/j.aim.2018.08.016
Y. Jiao, A. Osękowski and L. Wu, Noncommutative good-$\lambda$ inequalities, submitted
M. Junge and T. Mei, BMO spaces associated with semigroups of operators, Math. Ann. 352 (2012), no. 3, 691–743. MR 2885593, DOI 10.1007/s00208-011-0657-0
Marius Junge, Javier Parcet, and Quanhua Xu, Rosenthal type inequalities for free chaos, Ann. Probab. 35 (2007), no. 4, 1374–1437. MR 2330976, DOI 10.1214/009117906000000962
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
A. N. Kolmogorov, Sur les fonctions harmoniques conjugées et les séries de Fourier, Fund. Math. 7 (1925), 24–29.
Stanisław Kwapień and Wojbor A. Woyczyński, Random series and stochastic integrals: single and multiple, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1167198, DOI 10.1007/978-1-4612-0425-1
Tao Mei and Éric Ricard, Free Hilbert transforms, Duke Math. J. 166 (2017), no. 11, 2153–2182. MR 3694567, DOI 10.1215/00127094-2017-0007
Adam Osȩkowski, Sharp logarithmic inequalities for Riesz transforms, J. Funct. Anal. 263 (2012), no. 1, 89–108. MR 2920841, DOI 10.1016/j.jfa.2012.04.007
Adam Osękowski, Sharp inequalities for Riesz transforms, Studia Math. 222 (2014), no. 1, 1–18. MR 3215236, DOI 10.4064/sm222-1-1
Adam Osȩkowski, A sharp weak-type $(\infty ,\infty )$ inequality for the Hilbert transform, Complex Anal. Oper. Theory 10 (2016), no. 6, 1133–1143. MR 3532304, DOI 10.1007/s11785-015-0454-y
Narutaka Ozawa, A comment on free group factors, Noncommutative harmonic analysis with applications to probability II, Banach Center Publ., vol. 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 241–245. MR 2730894, DOI 10.4064/bc89-0-16
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
Gilles Pisier, Non-commutative vector valued $L_p$-spaces and completely $p$-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). MR 1648908
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
Marcel Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1928), no. 1, 218–244 (French). MR 1544909, DOI 10.1007/BF01171098
I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. of Math. (2) 39 (1938), no. 4, 811–841. MR 1503439, DOI 10.2307/1968466
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9