Sharp local lower 𝐿^{𝑝}-bounds for Dyadic-like maximal operators

Melas, Antonios; Nikolidakis, Eleftherios; Stavropoulos, Theodoros

doi:10.1090/S0002-9939-2013-11789-2

Sharp local lower $L^{p}$-bounds for Dyadic-like maximal operators
HTML articles powered by AMS MathViewer

by Antonios D. Melas, Eleftherios Nikolidakis and Theodoros Stavropoulos PDF

Proc. Amer. Math. Soc. 141 (2013), 3171-3181 Request permission

Abstract:

We provide sharp lower $L^{p}$-bounds for the localized dyadic maximal operator on $\mathbb {R}^{n}$ when the local $L^{1}$ and the local $L^{p}$ norm of the function are given. We actually do that in the more general context of homogeneous trees in probability spaces. For this we use an effective linearization 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
Antonios D. Melas, A sharp $L^p$ inequality for dyadic $A_1$ weights in $\Bbb R^n$, Bull. London Math. Soc. 37 (2005), no. 6, 919–926. MR 2186725, DOI 10.1112/S0024609305004765
Antonios D. Melas, Sharp general local estimates for dyadic-like maximal operators and related Bellman functions, Adv. Math. 220 (2009), no. 2, 367–426. MR 2466420, DOI 10.1016/j.aim.2008.09.010
Antonios D. Melas, Dyadic-like maximal operators on $L\log L$ functions, J. Funct. Anal. 257 (2009), no. 6, 1631–1654. MR 2540986, DOI 10.1016/j.jfa.2009.06.005
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
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
L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4135–4169. MR 2792983, DOI 10.1090/S0002-9947-2011-05112-3
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
E. M. Stein, Note on the class $L$ $\textrm {log}$ $L$, Studia Math. 32 (1969), 305–310. MR 247534, DOI 10.4064/sm-32-3-305-310
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
Vasily Vasyunin and Alexander Volberg, Monge-Ampère equation and Bellman optimization of Carleson embedding theorems, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 195–238. MR 2500520, DOI 10.1090/trans2/226/16
A. Volberg, Bellman approach to some problems in harmonic analysis, Seminaire des Equations aux derivées partielles, Ecole Polytéchnique, 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