Abstract
In this note we give the formula for the Bellman function associated with the problem considered by B. Davis (Duke Math J 43:697–704, 1976) in 1976. In this article the estimates of the type ∥Sf∥p ≤ C p∥f∥p, p ≥ 2, were considered for the dyadic square function operator S, and Davis found the sharp values of constants C p. However, along with the sharp constants one can consider a more subtle characteristic of the above estimate. This quantity is called the Bellman function of the problem. It has never been proved that the confluent hypergeometric function from Davis’ paper (second page) gives us this Bellman function. Here we fill out this gap by finding the exact Bellman function of the unweighted L p estimate for the dyadic square function operator S. We cast the proofs in the language of obstacle problems. For the sake of comparison, we also find the Bellman function of weak (1, 1) estimate of S. This formula was suggested by Bollobas (Math Proc Camb Phil Soc 87:377–382, 1980) and proved by Osekowski (Statist Probab Lett 79(13):1536–1538, 2009), so it is not new, but we like to emphasize the common approach to those two Bellman function descriptions.
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References
Barthe, F., Maurey, B.: Some remarks on isoperimetry of Gaussian type. Annales de l’Institut Henri Poincare (B) Probab. Stat. 36(4), 419–434 (2000)
Bollobás, B.: Martingale Inequalities. Math. Proc. Camb. Phil. Soc. 87, 377–382 (1980)
Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340(1), 253–272 (1993)
Burkholder, D.: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9, 997–1011 (1981)
Burkholder, D.: Boundary value problems and sharp estimates for the martingale transforms. Ann. Prob. 12, 647–702 (1984)
Burkholder, D.: Martingales and Fourier analysis in Banach spaces. In: Probability and Analysis (Varenna, 1985) Lecture Notes in Mathematical, vol. 1206, pp. 61–108 (1986)
Burkholder, D.: An extension of classical martingale inequality. In: Chao, J.-A., Woyczyński, W.A. (eds.) Probability Theory and Harmonic Analysis. Marcel Dekker, New York (1986)
Burkholder, D.: Sharp inequalities for martingales and stochastic integrals. In: Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque No. 157–158 (1988), pp. 75–94.
Burkholder, D.: A proof of the Pelczynski’s conjecture for the Haar system. Studia Math. 91, 79–83 (1988)
Burkholder, D.: Differential subordination of harmonic functions and martingales (El Escorial 1987). In: Lecture Notes in Mathematical, vol. 1384, pp. 1–23 (1989)
Burkholder, D.: Explorations of martingale theory and its applications. Lecture Notes in Math. 1464, 1–66 (1991)
Burkholder, D.: Strong differential subordination and stochastic integration. Ann. Prob. 22, 995–1025 (1994)
Burkholder, D.: Martingales and Singular Integrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. 1(Ch. 6), pp. 233–269 (2001)
Burkholder, D.L.: Martingale transforms. Ann. Math. Statist. 37, 1494–1504 (1966)
Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970)
Davis, B.: On the L p norms of stochastic integrals and other martingales. Duke Math. J. 43, 697–704 (1976)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions (1992)
Hall, R.R.: On a conjecture of Littlewood. Math. Proc. Cambridge Philos. Soc. 78, 443–445 (1975)
Holmes, I., Ivanisvili, P., Volberg, A.: The Sharp Constant in the Weak (1,1) Inequality for the Square Function: a new proof, Revista Mat. Iberoam. 36(3), 741–770 (2020)
Marcinkiewicz, J.: Quelque théorèmes sur les séries orthogonales. Ann. Soc. Polon. Math. 16, 84–96 (1937). (pages 307–318 of the Collected Papers)
Millar, P.W.: Martingale integrals. Trans. Am. Math. Soc. 133, 145–166 (1968)
Nazarov, F., Vasyunin, V., Volberg, A.L.: On Bellman function associated with Chang–Wilson–Wolff theorem. Algebra & Analysis, vol. 33, No. 4, pp. 66–106 (2021)
Novikov, A.A.: On stopping times for Wiener processes. Theory Probab. Appl. 16, 449–456 (1971)
Osekowski, A.: On the best constant in the weak type inequality for the square function of a conditionally symmetric martingale. Statist. Probab. Lett. 79(13), 1536–1538 (2009)
Osekowski, A.: Sharp martingale and semimartingale inequalities. In: Monografie Matematyczne, vol. 72. Springer, Basel (2012)
Osekowski, A.: Weighted square function inequalities. Publ. Mat. 62, 75–94 (2018)
Shepp, L.A.: A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914 (1967)
Szarek, S.J.: On the best constants in the Khintchine inequality. Studia Math. 18, 197–208 (1976)
Vasyunin, V., Volberg, A.: Bellman Function Technique in Harmonic Analysis (Cambridge University Press, Cambridge, 2020), pp. 1–460. https://doi.org/10.1017/9781108764469
Wang, G.: Some sharp inequalities for conditionally symmetric martingales, Ph.D. Thesis. University of Illinois, Urbana-Champaign (1989)
Wang, G.: Sharp square function inequalities for conditionally symmetric martingales. Trans. Am. Math. Soc. 328(1), 393–419 (1991)
Acknowledgements
I. Holmes is supported by National Science Foundation as an NSF Postdoc under Award No.1606270, A. Volberg is partially supported by the NSF DMS-1600065. This paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 A. Volberg was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring and Fall 2017 semester.
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Holmes, I., Volberg, A. (2021). Obstacle Problems Generated by the Estimates of Square Function. In: Ciatti, P., Martini, A. (eds) Geometric Aspects of Harmonic Analysis. Springer INdAM Series, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-030-72058-2_12
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