Let h R 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: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.

Here we show that Lerner's method of local mean oscillation gives a simple proof of the A 2 conjecture for spaces of homogeneous type: that is, the linear dependence on the A 2 norm for weighted L 2 Calderon-Zygmund operator estimates. In the Euclidean case, the result is due to Hytönen, and for geometrically doubling spaces, Nazarov, Rezinikov, and Volberg obtained the linear bound.

Abstract. Let A N be an N-point set in the unit square and consider the Discrepancy function, and |[ 0, x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of (Schmidt, 1972) The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we havewhenever N = 2 n for some positive integer n. This estimate depends upon variants of the Chang-Wilson-Wolff inequality (Chang et al., 1985). We also provide similar estimates for the BMO norm of D N .

Abstract. We recover one-dimensional Calderón-Zygmund convolution operators with sufficiently smooth kernels by means of a properly chosen averaging of certain dyadic shift operators. As a corollary, a sharp A 2 inequality for these Calderón-Zygmund operators is derived from a corresponding inequality for dyadic shift operators.