Abstract
The paper contains the study of sharp weighted logarithmic estimates for maximal operators on probability spaces equipped with a tree-like structure. These inequalities can be regarded as LlogL versions of the classical estimates of Fefferman and Stein. The proof exploits the existence of a certain special function, enjoying appropriate majorization and concavity conditions.
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References
Fefferman C., Stein E.M.: Some maximal inequalities. Amer. J. Math. 93, 107–115 (1971)
Gilat D.: The best bound in the LlogL inequality of Hardy and Littlewood and its martingale counterpart. Proc. Amer. Math. Soc. 97, 429–436 (1986)
Melas A. D.: The Bellman functions of dyadic-like maximal operators and related inequalities. Adv. Math. 192, 310–340 (2005)
Nazarov F., Treil S.: The hunt for Bellman function applications to estimates of singular integral operators and to other classical problems in harmonic analysis. Algebra i Analis 8, 32–162 (1997)
Shi X.: Two inequalities related to geometric mean operators. J. Zhejiang Teachers College 1, 21–25 (1980)
L. Slavin, A. Stokolos,and V. Vasyunin, Monge-Ampère equations and Bellman functions: The dyadic maximal operator, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 585–588.
Slavin L., Vasyunin V.: Sharp results in the integral-form John-Nirenberg inequality. Trans. Amer. Math. Soc. 363, 4135–4169 (2011)
L. Slavin and A. Volberg, Bellman function and the H 1-BMO duality, Harmonic analysis, partial differential equations, and related topics, 113–126, Contemp. Math., 428, Amer. Math. Soc., Providence, RI, 2007.
V. Vasyunin and A. Volberg, Monge-Ampère equation and Bellman optimization of Carleson embedding theorems, Linear and complex analysis, pp. 195–238, Amer. Math. Soc. Transl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009.
Vasyunin V., Volberg A.: Burkholder’s function via Monge-Ampère equation. Illinois J. Math. 54, 1393–1428 (2010)
Wittwer J.: Survey article: a user’s guide to Bellman functions. Rocky Mountain J. Math. 41, 631–661 (2011)
Yin X., Muckenhoupt B.: Weighted inequalities for the maximal geometric mean operator. Proc. Amer. Math. Soc. 124, 75–81 (1996)
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Research supported by the National Science Center, Poland, Grant DEC-2014/14/E/ST1/00532.
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Osȩkowski, A. Sharp weighted logarithmic bound for maximal operators. Arch. Math. 107, 635–644 (2016). https://doi.org/10.1007/s00013-016-0980-5
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DOI: https://doi.org/10.1007/s00013-016-0980-5