TY - JOUR
T1 - On the Small Ball Inequality in all dimensions
AU - Bilyk, Dmitriy
AU - Lacey, Michael T.
AU - Vagharshakyan, Armen
PY - 2008/5/1
Y1 - 2008/5/1
N2 - Let hR denote an L∞ normalized Haar function adapted to a dyadic rectangle R ⊂ [0, 1]d. We show that for choices of coefficients α (R), we have the following lower bound on the L∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:nfrac(d - 1, 2) - η {norm of matrix} under(∑, | R | = 2- n) α (R) hR (x) {norm of matrix}L∞ ([0, 1]d) ≳ 2- n under(∑, | R | = 2- n) | α (R) |, for some 0 < η < frac(1, 2) . The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n(d - 1) / 2 on the left above, the inequality is trivial, while it is conjectured that the inequality holds with n(d - 2) / 2. This is known in the case of d = 2 [Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (3) (1994) 1331-1354, MR 95k:60049], and a recent paper of two of the authors [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] proves a partial result towards the conjecture in three dimensions. In this paper, we show that the argument of [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] can be extended to arbitrary dimension. We also prove related results in the subjects of the irregularity of distribution, and approximation theory. The authors are unaware of any prior results on these questions in any dimension d ≥ 4.
AB - Let hR denote an L∞ normalized Haar function adapted to a dyadic rectangle R ⊂ [0, 1]d. We show that for choices of coefficients α (R), we have the following lower bound on the L∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:nfrac(d - 1, 2) - η {norm of matrix} under(∑, | R | = 2- n) α (R) hR (x) {norm of matrix}L∞ ([0, 1]d) ≳ 2- n under(∑, | R | = 2- n) | α (R) |, for some 0 < η < frac(1, 2) . The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n(d - 1) / 2 on the left above, the inequality is trivial, while it is conjectured that the inequality holds with n(d - 2) / 2. This is known in the case of d = 2 [Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (3) (1994) 1331-1354, MR 95k:60049], and a recent paper of two of the authors [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] proves a partial result towards the conjecture in three dimensions. In this paper, we show that the argument of [Dmitriy Bilyk, Michael T. Lacey, On the Small Ball Inequality in three dimensions, Duke Math. J., (2006), in press, arXiv: math.CA/0609815] can be extended to arbitrary dimension. We also prove related results in the subjects of the irregularity of distribution, and approximation theory. The authors are unaware of any prior results on these questions in any dimension d ≥ 4.
KW - Brownian Sheet
KW - Discrepancy function
KW - Haar functions
KW - Kolmogorov entropy
KW - Littlewood-Paley Inequalities
KW - Mixed derivative
KW - Small ball inequality
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U2 - 10.1016/j.jfa.2007.09.010
DO - 10.1016/j.jfa.2007.09.010
M3 - Article
AN - SCOPUS:41049098063
SN - 0022-1236
VL - 254
SP - 2470
EP - 2502
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 9
ER -