Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Antonios D. Melas; Eleftherios Nikolidakis
Journal: Trans. Amer. Math. Soc. 362 (2010), 1571-1597.
MSC (2000): Primary 42B25
Posted: October 20, 2009
MathSciNet review: 2563741
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: For each 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:

1.: D. L. Burkholder, Martingales and Fourier analysis in Banach spaces, C.I.M.E. Lectures (Varenna (Como), Italy, 1985), Lecture Notes in Mathematics 1206 (1986), 61-108. MR 864712 (88c:42017)
2.: D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. of Prob. 12 (1984), 647-702. MR 744226 (86b:60080)
3.: L. Grafakos, S. Montgomery-Smith. Best constants for uncentered maximal functions, Bull. London Math. Soc. 29 (1997), no.1, 60-64. MR 1416408 (98b:42031)
4.: A. D. Melas, The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. in Math. 192 (2005), 310-340. MR 2128702 (2005k:42052)
5.: F. Nazarov, S. Treil, 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 no. 5 (1996), 32-162. MR 1428988 (99d:42026)
6.: F. Nazarov, S. Treil, A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers. Journ. Amer. Math. Soc. 12 no. 4 (1999), 909-928. MR 1685781 (2000k:42009)
7.: F. Nazarov, S. Treil, A. Volberg, Bellman function in stochastic optimal control and harmonic analysis (how our Bellman function got its name), Oper. Theory: Advances and Appl. 129 (2001), 393-424, Birkhäuser Verlag. MR 1882704 (2003b:49024)
8.: L. Slavin, V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, submitted.
9.: L. Slavin, A. Stokolos, V. Vasyunin. Monge-Ampère equations and Bellman functions: The dyadic maximal operator. C. R. Math. Acad. Sci. Paris Sér. I, 346 (2008), 585-588. MR 2412802
10.: L. Slavin, A. Volberg. The explicit BF for a dyadic Chang-Wilson-Wolff theorem. The s-function and the exponential integral. Contemp. Math. 444, Amer. Math. Soc., Providence, RI, 2007. MR 2423630
11.: E. M. Stein. Harmonic Analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993. MR 1232192 (95c:42002)
12.: V. Vasyunin. The sharp constant in the reverse Holder inequality for Muckenhoupt weights. Algebra i Analiz, 15 (2003), no. 1, 73-117 MR 1979718 (2004h:42017)
13.: V. Vasyunin, A. Volberg. The Bellman functions for the simplest two-weight inequality: The case study. Algebra i Analiz, 18 (2006), No. 2 MR 2244935 (2007k:47053)
14.: V. Vasyunin, A. Volberg. Monge-Ampère equation and Bellman optimization of Carleson embedding Theorem, preprint.
15.: A. Volberg, Bellman approach to some problems in Harmonic Analysis, Séminaire des Equations aux derivées partielles, Ecole Polytechnique, 2002, exposé. XX, 1-14.
16.: G. Wang, Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion, Proc. Amer. Math. Soc. 112 (1991), 579-586. MR 1059638 (91i:60121)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|