Estimates for discrepancy and Calderon-Zygmund operators

The thesis consists of two independent parts.The first part is devoted to certain results in the discrepancy theory and related problems. Take A &sub [0, 1] d to be an N point set in the d dimensional unit cube and consider the discrepancy function associated to it: DAx&ar :=&sharp&cubl0A&cap&sqbl00&ar ,x&ar&sqbr0&cubr0-N&sqbl0 0&ar,x&ar &sqbr0,x&ar &isin0,1d. (here the &sharp sign counts the number of elements in the set, and &sqbl00&ar,x&ar &sqbr0 stands for the rectangle with antipodal corners 0&ar and x&ar ). The function DA measures how much the distribution of the finite set A deviates from the corresponding uniform distribution.In a joint work with D. Bilyk and M. Lacey we extended the previous result of D. Bilyk and M. Lacey (see [4]) to dimensions d > 3, by improving the lower bound for the discrepancy function. Namely, we showed that there exists a positive eta(d) > 0 for which we have: DA infinity&gsimln Nd-1 /2+hd foranyset &sharpA=N.This result makes a partial step towards resolving the Discrepancy Conjecture. Being a theorem in the theory of irregularities of distributions, it also relates to corresponding results in approximation theory (namely, the Kolmogorov entropy of spaces of functions with bounded mixed derivatives) and in probability theory (namely, Small Ball Inequality - small deviation inequality for the Brownian sheet).In another joint work with D. Bilyk and M. Lacey we treat a particular case of the Small Ball Inequality - the Signed Small Ball Inequality. We show that in this case our estimates can be further improved.Yet another joint work with D. Bilyk, M. Lacey and I. Parissis provides sharp bounds for the exponential Orlicz norm and the BMO norm of the discrepancy function in two dimensions.The second part of the thesis deals with Calderon-Zygmund operators in weighted spaces. We prove that any sufficiently smooth one-dimensional Calderon-Zygmund convolution operator can be recovered through averaging of certain Haar shift operators (i.e. dyadic operators which can be efficiently expressed in terms of the Haar basis). This generalizes the estimates, which had been previously known (see [23]) for Haar shift operators, to Calderon-Zygmund operators. As a result, the A2 conjecture is settled for this particular type of Calderon-Zygmund operators.