Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov’s inequality

Melas, Antonios; Nikolidakis, Eleftherios

doi:10.1090/S0002-9947-09-04872-7

Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov’s inequality
HTML articles powered by AMS MathViewer

by Antonios D. Melas and Eleftherios Nikolidakis PDF

Trans. Amer. Math. Soc. 362 (2010), 1571-1597 Request permission

Abstract:

For each $q<1$ we precisely evaluate the main Bellman functions associated with the behavior of dyadic maximal operators on $\mathbb {R}^{n}$ on integrable functions. Actually we do that in the more general setting of tree-like maximal operators. These are related to and refine the corresponding Kolmogorov inequality, which we show is actually sharp. For this we use the effective linearization introduced by the first author in 2005 for such maximal operators on an adequate set of functions.

References

Donald L. Burkholder, Martingales and Fourier analysis in Banach spaces, Probability and analysis (Varenna, 1985) Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, pp. 61–108. MR 864712, DOI 10.1007/BFb0076300
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
Loukas Grafakos and Stephen Montgomery-Smith, Best constants for uncentred maximal functions, Bull. London Math. Soc. 29 (1997), no. 1, 60–64. MR 1416408, DOI 10.1112/S0024609396002081
Antonios D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math. 192 (2005), no. 2, 310–340. MR 2128702, DOI 10.1016/j.aim.2004.04.013
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
F. Nazarov, S. Treil, and A. Volberg, Bellman function in stochastic control and harmonic analysis, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 393–423. MR 1882704
L. Slavin, V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, submitted.
Leonid Slavin, Alexander Stokolos, and Vasily Vasyunin, Monge-Ampère equations and Bellman functions: the dyadic maximal operator, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 585–588 (English, with English and French summaries). MR 2412802, DOI 10.1016/j.crma.2008.03.003
Leonid Slavin and Alexander Volberg, The $s$-function and the exponential integral, Topics in harmonic analysis and ergodic theory, Contemp. Math., vol. 444, Amer. Math. Soc., Providence, RI, 2007, pp. 215–228. MR 2423630, DOI 10.1090/conm/444/08582
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
V. I. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Algebra i Analiz 15 (2003), no. 1, 73–117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 1, 49–79. MR 1979718, DOI 10.1090/S1061-0022-03-00802-1
V. Vasyunin and A. Vol′berg, The Bellman function for the simplest two-weight inequality: an investigation of a particular case, Algebra i Analiz 18 (2006), no. 2, 24–56 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 2, 201–222. MR 2244935, DOI 10.1090/S1061-0022-07-00953-3
V. Vasyunin, A. Volberg. Monge-Ampère equation and Bellman optimization of Carleson embedding Theorem, preprint.
A. Volberg, Bellman approach to some problems in Harmonic Analysis, Séminaire des Equations aux derivées partielles, Ecole Polytechnique, 2002, exposé. XX, 1-14.
Gang Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991), no. 2, 579–586. MR 1059638, DOI 10.1090/S0002-9939-1991-1059638-8