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SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

Part of: Harmonic analysis in several variables Linear function spaces and their duals Stochastic processes

Published online by Cambridge University Press:  10 June 2016

ADAM OSȨKOWSKI
ADAM OSȨKOWSKI*
Affiliation:
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland e-mail: [email protected]
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Abstract

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Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$.

  1. (i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate

    $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$
    For each p, the constant e1/p is the best possible.

  2. (ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$,

    $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$
    and prove that the constant e is optimal.

Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 60G42: Martingales with discrete parameter
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 533 - 547
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Cruz-Uribe, D., The minimal operator and the geometric maximal operator in $\mathbb{R}^n$ , Studia Math. 144 (1) (2001), 1–37.CrossRefGoogle Scholar
2. Cruz-Uribe, D. and Neugebauer, C. J., Weighted norm inequalities for the geometric maximal operator, Publ. Mat. 42 (1) (1998), 239–263.CrossRefGoogle Scholar
3. Fefferman, C. and Stein, E. M., Some maximal inequalities, Amer. J. Math. 93 (1) (1971), 107–115.CrossRefGoogle Scholar
4. Melas, A. D., The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math. 192 (2) (2005), 310–340.CrossRefGoogle Scholar
5. Nazarov, F. and 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 (5) (1997), 32–162.Google Scholar
6. Ortega Salvador, P. and Ramírez Torreblanca, C., Weighted inequalities for the one-sided geometric maximal operators, Math. Nachr. 284 (11–12) (2011), 1515–1522.CrossRefGoogle Scholar
7. Shi, X., Two inequalities related to geometric mean operators, J. Zhejiang Teachers College 1 (1980), 21–25.Google Scholar
8. Slavin, L., Stokolos, A. and Vasyunin, V., Monge-Ampère equations and Bellman functions: The dyadic maximal operator, C. R. Acad. Sci. Paris, Ser. I 346 (9–10) (2008), 585–588.CrossRefGoogle Scholar
9. Slavin, L. and Vasyunin, V., Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (8) (2011), 4135–4169.CrossRefGoogle Scholar
10. Slavin, L. and Volberg, A., Bellman function and the H 1-BMO duality, Harmonic analysis, partial differential equations, and related topics, Contemp. Math., vol. 428 (American Mathematical Society, Providence, RI, 2007), 113–126.CrossRefGoogle Scholar
11. Vasyunin, V. and Volberg, A., Monge-Ampére equation and Bellman optimization of Carleson embedding theorems, Linear and complex analysis, Amer. Math. Soc. Transl. Ser., vol. 2, 226 (American Mathematical Society, Providence, RI, 2009), pp. 195–238.Google Scholar
12. Vasyunin, V. and Volberg, A., Burkholder's function via Monge-Ampére equation, Illinois J. Math. 54 (4) (2010), 1393–1428.CrossRefGoogle Scholar
13. Wittwer, J., Survey article: A user's guide to Bellman functions, Rocky Mountain J. Math. 41 (3) (2011), 631–661.CrossRefGoogle Scholar
14. Yin, X. and Muckenhoupt, B., Weighted inequalities for the maximal geometric mean operator, Proc. Amer. Math. Soc. 124 (1) (1996), 75–81.CrossRefGoogle Scholar