We present lower estimates for the best constant appearing in the weak (1, 1) maximal inequality in the space (Rn, ‖ · ‖∞). We show that this constant grows to infinity faster than (log n)1-o(1) when n tends to infinity. To this end, we follow and simplify the approach used by J. M. Aldaz. The new part of the argument relies on Donsker's theorem identifying the Brownian bridge as the limit object describing the statistical distribution of the coordinates of a point randomly chosen in the unit cube [0, 1]n (n large).
@article{CML_2009__1_2_169_0,
author = {Aubrun, Guillaume},
title = {Maximal inequality for high-dimensional cubes},
journal = {Confluentes Mathematici},
pages = {169--179},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {1},
number = {2},
year = {2009},
doi = {10.1142/S1793744209000067},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744209000067/}
}
TY - JOUR AU - Aubrun, Guillaume TI - Maximal inequality for high-dimensional cubes JO - Confluentes Mathematici PY - 2009 SP - 169 EP - 179 VL - 1 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744209000067/ DO - 10.1142/S1793744209000067 LA - en ID - CML_2009__1_2_169_0 ER -
%0 Journal Article %A Aubrun, Guillaume %T Maximal inequality for high-dimensional cubes %J Confluentes Mathematici %D 2009 %P 169-179 %V 1 %N 2 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744209000067/ %R 10.1142/S1793744209000067 %G en %F CML_2009__1_2_169_0
Aubrun, Guillaume. Maximal inequality for high-dimensional cubes. Confluentes Mathematici, Tome 1 (2009) no. 2, pp. 169-179. doi : 10.1142/S1793744209000067. http://www.numdam.org/articles/10.1142/S1793744209000067/
[1] J. M. Aldaz, The weak type (1,1) bounds for the maximal function associated to cubes grow to infinity with the dimension , arXiv:0805.1565 .
[2] J. Bourgain, Israel J. Math. 54, 257 (1986), DOI: 10.1007/BF02764955 .
[3] J. Bretagnolle and P. Massart, Ann. Probab. 17, 239 (1989), DOI: 10.1214/aop/1176991506 .
[4] A. Carbery, Bull. Amer. Math. Soc. (N.S.) 14, 269 (1986), DOI: 10.1090/S0273-0979-1986-15436-4 .
[5] A. Criado and F. Soria, On the growth with respect to the dimension of the weak type constant for the centered maximal operator associated with cubes, preprint .
[6] M. D. Donsker, Ann. Math. Statist. 23, 277 (1952), DOI: 10.1214/aoms/1177729445 .
[7] R. Durrett , Probability: Theory and Examples , 3rd edn. ( Duxbury Press , 1996 ) .
[8] J. Komlós, P. Major and G. Tusnády, Z. Wahrsch. Verw. Gebiete 32, 111 (1975).
[9] W. Rudin , shape Real and Complex Analysis , 3rd edn. ( McGraw-Hill , 1987 ) .
[10] E. M. Stein and J.-O. Strömberg, Ark. Mat. 21, 259 (1983), DOI: 10.1007/BF02384314 .
Cité par Sources :
