# zbMATH — the first resource for mathematics

Subordination by conformal martingales in $$L^{p}$$ and zeros of Laguerre polynomials. (English) Zbl 1266.32006
Summary: Given martingales $$W$$ and $$Z$$ such that $$W$$ is differentially subordinate to $$Z$$, D. Burkholder obtained the sharp inequality $$\mathbb E|W|^{p}\leq(p^{*}-1)^{p}\mathbb E|Z|^{p}$$, where $p^{*}=\max\Big\{p,\frac{p}{p-1}\Big\}.$ What happens if one of the martingales is also a conformal martingale? R. Bañuelos and P. Janakiraman [Trans. Am. Math. Soc. 360, No. 7, 3603–3612 (2008; Zbl 1220.42012)] proved that if $$p\geq2$$ and $$W$$ is a conformal martingale differentially subordinate to any martingale $$Z$$, then $\mathbb E|W|^{p}\leq \bigg [\frac{p^{2}-p}{2}\bigg]^{p/2}\mathbb E|Z|^{p}.$
In this paper, we establish that if $$p\geq2$$, $$Z$$ is conformal, and $$W$$ is any martingale subordinate to $$Z$$, then $\mathbb E|W|^{p}\leq\bigg[\sqrt{2}\frac{1-z_{p}}{z_{p}}\bigg]^{p}\mathbb E|Z|^{p},$ where $$z_{p}$$ is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for $$1< p < 2$$. Finally, we give an application of our results. Previous estimates on the $$L^{p}$$-norm of the Beurling-Ahlfors transform give at best $$\|B\|_{p}\lesssim\sqrt{2}p$$ as $$p\rightarrow\infty$$. We improve this to $$\|B\|_{p}\lesssim1.3922\,p$$ as $$p\rightarrow\infty$$.

##### MSC:
 32A55 Singular integrals of functions in several complex variables 60G46 Martingales and classical analysis 42A15 Trigonometric interpolation 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Full Text:
##### References:
 [1] K. Astala, Area distortion of quasiconformal mappings , Acta Math. 173 (1994), 37-60. · Zbl 0815.30015 [2] A. Baernstein and S. Montgomry-Smith, “Some conjectures about integral means of $$\partial f$$ and $$\overline{\partial}f$$” in Complex Analysis and Differential Equations , Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. 64 , Uppsala Univ., Uppsala, 1999, 92-109. · Zbl 0966.30001 [3] R. Bañuelos, The foundational inequalities of D.L. Burkholder and some of their ramifications , Illinois J. Math. 54 (2010), 789-868. · Zbl 1259.60047 [4] R. Bañuelos and P. Janakiraman, $$L^{p}$$-bounds for the Beurling-Ahlfors transform , Trans. Amer. Math. Soc. 360 (2008), 3603-3612. · Zbl 1220.42012 [5] R. Bañuelos and P. J. Méndez-Hernández, Space-time Brownian motion and the Beurling-Ahlfors transform , Indiana Univ. Math. J. 52 (2003), 981-990. · Zbl 1080.60043 [6] R. Bañuelos and A. Ose\ogonek kowski, Burkholder inequalities for submartingales, Bessel processes and conformal martingales , preprint, [math.PR]. 1112.0551v1 · Zbl 1286.60041 [7] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms , Duke Math. J. 80 (1995), 575-600. · Zbl 0853.60040 [8] A. Borichev, P. Janakiraman, A. Volberg, On Burkholder function for orthogonal martingales and zeros of Legendre polynomials , Amer. J. Math. 135 (2013), 207-236. · Zbl 1266.60079 [9] D. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional , Ann. Probab. 9 (1981), 997-1011. · Zbl 0474.60036 [10] D. Burkholder, Boundary value problems and sharp estimates for the martingale transforms , Ann. Probab. 12 (1984), 647-702. · Zbl 0556.60021 [11] D. Burkholder, “Martingales and Fourier analysis in Banach spaces” in Probability and Analysis (Varenna, 1985) , Lecture Notes in Math. 1206 , Springer, Berlin, 1986, 61-108. · Zbl 0605.60049 [12] D. Burkholder, “An extension of classical martingale inequality” in Probability Theory and Harmonic Analysis (Cleveland, Ohio, 1983) . Monogr. Textbooks Pure Appl. Math. 98 , Dekker, New York, 1986, 21-30. · Zbl 0594.60020 [13] D. Burkholder, “Sharp inequalities for martingales and stochastic integrals” in Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987) , Astérisque No. 157-158, 1988, 75-94. · Zbl 0656.60055 [14] D. Burkholder, A proof of the Peczynski’s conjecture for the Haar system , Studia Math. 91 (1988), 79-83. · Zbl 0652.42012 [15] D. Burkholder, “Differential subordination of harmonic functions and martingales” in Harmonic Analysis and Partial Differential Equations (El Escorial 1987) , Lecture Notes in Math. 1384 , 1989, 1-23. · Zbl 0675.31003 [16] D. Burkholder, “Explorations of martingale theory and its applications” in École d’Été de Probabilités de Saint-Flour XIX-1989 , Lecture Notes in Math. 1464 , 1991, 1-66. · Zbl 0771.60033 [17] D. Burkholder, Strong differential subordination and stochastic integration , Ann. Prob. 22 (1994), 995-1025. · Zbl 0816.60046 [18] D. Burkholder, “Martingales and singular integrals in Banach spaces” in Handbook of the Geometry of Banach Spaces , Vol. 1, North-Holland, Amsterdam, 2001, 233-269. · Zbl 1029.46007 [19] E. A. Coddington, An Introduction to Ordinary Differential Operators , New York, Dover, 1989. [20] E. A. Coddington and N. Levinson, The Theory of Ordinary Differential Operators , McGraw-Hill, New York, 1955. · Zbl 0064.33002 [21] O. Dragičević, S. Treil, and A. Volberg, A theorem about $$3$$ quadratic forms , Int. Math. Res. Not. IMRN. 2008, Art. ID rnn072. · Zbl 1245.42012 [22] O. Dragičević and A. Volberg, Sharp estimates of the Ahlfors-Beurling operator via averaging of martingale transform , Michigan Math. J. 51 (2003), 415-435. · Zbl 1056.42011 [23] O. Dragičević and A. Volberg, Bellman function, Littlewood-Paley estimates, and asymptotics of the Ahlfors-Beurling operator in $$L^{p}(\mathbb{C})$$, $$p\rightarrow\infty$$ , Indiana Univ. Math. J. 54 (2005), 971-995. · Zbl 1103.47036 [24] J. Duoandikoetxea, Fourier Analysis , Grad. Stud. in Math. 29 , Amer. Math. Soc., Providence, 2001. · Zbl 0969.42001 [25] S. Geiss, S. Montgomery-Smith, and E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362 (2010), 553-575. · Zbl 1196.60078 [26] T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings , Z. Anal. Anwendungen 1 (1982), 1-16. · Zbl 0577.46038 [27] P. Janakiraman, Orthogonality complex in martingale spaces and connections with the Beurling-Ahlfors transform , Illinois J. Math. 54 (2010), 1509-1563. · Zbl 1259.60046 [28] I. Karatzas and S. Shreve, Brownian motion and stochastic calculus , Grad. Texts in Math., Springer, New York, 1991. · Zbl 0734.60060 [29] N. Krylov, Controlled Diffusion Processes , ed. 2, Appli. Math. 14 , Springer, New York, 2008. [30] O. Lehto, Remarks on the integrability of the derivatives of quasiconformal mappings , Ann. Acad. Sci. Fenn. Series A I Math. 371 (1965), 8 pp. · Zbl 0137.05503 [31] F. Nazarov and S. Treil, The hunt for a Bellman function: Applications to estimates of singular integral operators and to other classical problems in harmonic analysis , St. Petersburg Math. J. 8 (1997), 721-824. [32] F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers , J. Amer. Math. Soc. 12 (1999), 909-928. · Zbl 0951.42007 [33] F. Nazarov, S. Treil, A. Volberg, “Bellman function in stochastic control and harmonic analysis” in Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000) , Oper. Theory Adv. Appl. 129 , Birkhauser, Basel, 2001, 393-423. · Zbl 0999.60064 [34] F. Nazarov and A. Volberg, Bellman function, two weighted Hilbert transforms and embeddings of the model spaces $$K_{\theta}$$ , J. Anal. Math. 87 (2002), 385-414. · Zbl 1035.42010 [35] F. Nazarov, A. Volberg, Heating of the Ahlfors-Beurling operator and estimates of its norm , St. Petersburg Math. J. 15 (2004), 563-573. · Zbl 1061.47042 [36] S. Petermichl, A sharp bound for weighted Hilbert transform in terms of classical $$A_{p}$$ characteristic , Amer. J. Math. 129 (2007), 1355-1375. · Zbl 1139.44002 [37] S. Petermichl and A. Volberg, Heating the Beurling operator: Weakly quasiregular maps on the plane are quasiregular , Duke Math. J. 112 (2002), 281-305. · Zbl 1025.30018 [38] S. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund, and Kolmogorov , Studia Math. 44 (1972), 165-179. · Zbl 0238.42007 [39] A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces , Transl. Math. Monogr. 35 , Amer. Math. Soc., Providence, 1973. · Zbl 0311.53067 [40] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol 1 , Cambridge Univ. Press, Cambridge, 2000. · Zbl 0977.60005 [41] L. Slavin and A. Stokolos, The Bellman PDE for the dyadic maximal function and its solution , preprint, 2006. [42] L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality , Trans. Amer. Math. Soc. 363 (2011), 4135-4169. · Zbl 1223.42001 [43] G. Szegö, Orthogonal Polynomials , 4th ed., Amer. Math. Soc., Colloq. Publ. 23 , Amer. Math. Soc., Providence, 1975. [44] V. Vasyunin, and A. Volberg, The Bellman function for certain two weight inequality: The case study , St. Petersburg Math. J. 18 (2007) 201-222. · Zbl 1125.47025 [45] V. Vasyunin, A. Volberg, “Monge-Ampère equation and Bellman optimization of Carleson embedding theorems” in Linear and Complex Analysis , Amer. Math. Soc. Transl. Ser. 2, 226 , Amer. Math. Soc., Providence, 2009, 195-238. · Zbl 1178.28017 [46] V. Vasyunin, A. Volberg, Bellman Functions Technique in Harmonic Analysis , preprint, 2009, pp. 1-86. [47] A. Volberg, Bellman approach to some problems in Harmonic Analysis , Séminaires des Equations aux derivées partielles, Ecole Politéchnique, 2002, exposé XX, pp. 1-14. [48] G. N. Watson, A treatise on the theory of Bessel functions , reprint of the 1994 second edition, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.