Localization and dimension free estimates for maximal functions. (English) Zbl 1287.42013
A. Naor and T. Tao [J. Funct. Anal. 259, No. 3, 731–779 (2010; Zbl 1196.42018)] recently introduced a new class of measures with a so-called micro-doubling property and proved a localization principle for the associated maximal operators. This proof by Naor and Tao is probabilistic and relies on random martingales and Dood-type maximal inequalities. Consequently, Naor and Tao observed that in order to extend Stein’s and Strömberg’s ’\(n\log n\)’ bound to general measure spaces it suffices to assume that dilations by the factor \(\displaystyle \left(1+\frac{1}{n}\right)\) preserve essentially the volumes of balls and that the measures of intersecting balls with the same radius are comparable. The aim of the present paper is twofold. First, the two authors prove a new localization principle that considers localization not only in time but also in space. The proof is geometrical and is based on covering lemmas and selection processes. This principle may be used to prove the maximal theorem presented by Naor and Tao. Second, the authors restrict themselves to the Euclidean case and prove that a uniform condition for micro-doubling measures provides dimension free bounds for their maximal functions in all \(\displaystyle \mathbb{L}^p\), for \(p>1\). This is achieved by means of a new technique that allows differentiation through dimensions. In the last section, the authors provide some examples of doubling measures with associated maximal operators that fail to have uniform bounds.
Reviewer: Anna Savvopoulou (South Bend)
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
Citations:
Zbl 1196.42018References:
| [1] | Aldaz, J. M., Dimension dependency of the weak type \((1, 1)\) bounds for maximal functions associated to finite radial measures, Bull. Lond. Math. Soc., 39, 203-208 (2007) · Zbl 1133.42026 |
| [2] | Aldaz, J. M., The weak type \((1, 1)\) bounds for the maximal function associated to cubes grow to infinity with the dimension, Ann. of Math. (2), 173, 2, 1013-1023 (2011) · Zbl 1230.42025 |
| [3] | Aldaz, J. M.; Pérez Lázaro, J., Dimension dependency of \(L^p\) bounds for maximal functions associated to radial measures, Positivity, 15, 199-213 (2011) · Zbl 1221.42031 |
| [4] | Aldaz, J. M.; Pérez Lázaro, J., On high dimensional maximal operators, Banach J. Math. Anal., 7, 2, 225-243 (2013) · Zbl 1266.42041 |
| [5] | Aubrun, G., Maximal inequality for high-dimensional cubes, Confluentes Math., 1, 2, 169-179 (2009) · Zbl 1177.42015 |
| [6] | Bourgain, J., On high-dimensional maximal function associated to convex bodies, Amer. J. Math., 108, 6, 1467-1476 (1986) · Zbl 0621.42015 |
| [7] | Bourgain, J., On the \(L^p\)-bounds for maximal functions associated to convex bodies in \(R^n\), Israel J. Math., 54, 3, 257-265 (1986) · Zbl 0616.42013 |
| [8] | Bourgain, J., On dimension free maximal inequalities for convex symmetric bodies in \(R^n\), (Geometrical Aspects of Functional Analysis, Israel Seminar 1985-86. Geometrical Aspects of Functional Analysis, Israel Seminar 1985-86, Lecture Notes in Math., vol. 1267 (1987), Springer: Springer Berlin), 168-176 · Zbl 0622.52005 |
| [9] | Bourgain, J., On the Hardy-Littlewood maximal function for the cube · Zbl 1309.42023 |
| [10] | Carbery, A., An almost-orthogonality principle with applications to maximal functions associated to convex bodies, Bull. Amer. Math. Soc. (N.S.), 14, 2, 269-273 (1986) · Zbl 0588.42012 |
| [11] | Criado, A., On the lack of dimension free estimates in \(L^p\) for maximal functions associated to radial measures, Proc. Roy. Soc. Edinburgh Sect. A, 140, 3, 541-552 (2010) · Zbl 1210.42034 |
| [12] | Criado, A., Problems of harmonic analysis in high dimensions (2012), Universidad Autónoma de Madrid, Ph.D. thesis |
| [13] | Criado, A.; Sjögren, P., Bounds for maximal functions associated with rotational invariant measures in high dimensions, J. Geom. Anal. (2012) |
| [14] | de Guzmán, M., Differentiation of Integrals in \(R^n\), Lecture Notes in Math., vol. 481 (1975), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0327.26010 |
| [15] | Duoandikoetxea, J.; Vega, L., Spherical means and weighted inequalities, J. Lond. Math. Soc. (2), 53, 2, 343-353 (1996) · Zbl 0853.42017 |
| [16] | García-Cuerva, J.; Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics, Notas Mat., vol. 104 (1985), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam, North-Holland Math. Stud., vol. 116 · Zbl 0578.46046 |
| [17] | Garsia, A. M., Topics in Almost Everywhere Convergence, Lect. Adv. Math., vol. 4 (1970), Markham Publishing Co.: Markham Publishing Co. Chicago, IL · Zbl 0198.38401 |
| [18] | Infante, A., Análisis armónico para medidas no doblantes: operadores maximales con respecto a la medida gaussiana (2005), Universidad Autónoma de Madrid, Ph.D. thesis |
| [19] | Infante, A., Free-dimensional boundedness of the maximal operator, Bol. Soc. Mat. Mexicana (3), 14, 1, 67-73 (2008) · Zbl 1202.42040 |
| [20] | Lindenstrauss, E., Pointwise theorems for amenable groups, Invent. Math., 146, 2, 259-295 (2001) · Zbl 1038.37004 |
| [21] | Menárguez, M. T., Técnicas de discretización en Análisis Armónico para el estudio de acotaciones débiles de operadores maximales e integrales singulares (1990), Universidad Complutense: Universidad Complutense Madrid, Ph.D. thesis |
| [22] | Menárguez, M. T.; Soria, F., Weak type \((1, 1)\) inequalities for maximal convolution operators, Rend. Circ. Mat. Palermo (2), 41, 3, 342-352 (1992) · Zbl 0770.42013 |
| [23] | Menárguez, M. T.; Soria, F., On the maximal operator associated to a convex body in \(R^n\), Collect. Math., 3, 243-251 (1992) · Zbl 0792.42007 |
| [24] | Muckenhoupt, B.; Stein, E. M., Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118, 17-92 (1965) · Zbl 0139.29002 |
| [25] | Müller, D., A geometric bound for maximal functions associated to convex bodies, Pacific J. Math., 142, 2, 297-312 (1990) · Zbl 0728.42015 |
| [26] | Naor, A.; Tao, T., Random martingales and localization of maximal inequalities, J. Funct. Anal., 259, 3, 731-779 (2010) · Zbl 1196.42018 |
| [27] | Stein, E. M., The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.), 7, 2, 359-376 (1982) · Zbl 0526.01021 |
| [28] | Stein, E. M.; Strömberg, J. O., Behavior of maximal functions in \(R^n\) for large \(n\), Ark. Mat., 21, 2, 250-269 (1983) · Zbl 0537.42018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
