On the Discrepancy Function in Arbitary Dimension, Close to L ^{1}

Let $\mathcal A_N$ to be $N$ points in the unit cube in dimension $d$, and consider the Discrepency function D_N(\vec x) \coloneqq \sharp \mathcal A_N \cap [\vec 0,\vec x)-N \abs{[\vec 0,\vec x)} Here, $\vec x= (x_1 ,...c, x_d)$ and $[ 0,\vec x)=\prod_{t=1} ^{d} [0,x_t)$. We show that necessarily \norm D_N. L ^{1} (\log L) ^{(d-2)/2}. \gtrsim (\log N) ^{d/2} . In dimension $d=2$, the `$\log L$' term has power zero, which corresponds to a Theorem due to \cite{MR637361}.Comment: 17 pages. To appear in Analysis Mathematica. Many changes, and an additional section on Hardy space and the Discrepancy functio

Commutators with Reisz Potentials in One and Several Parameters

Let $M_b$ be the operator of pointwise multiplication by $b$, that is $\operatorname M_b f=bf$. Set $[ A,B]={} AB- BA$. The Reisz potentials are the operators R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha}},\qquad 0<\alpha<1. They map $L^p\mapsto L^q$, for $1-\alpha+\frac1q=\frac1p$, a fact we shall take for granted in this paper. A Theorem of Chanillo \cite{MR84j:42027} states that one has the equivalence \norm [ M_b, R_\alpha].p\to q.\simeq \norm b.\operatorname{BMO}. with the later norm being that of the space of functions of bounded mean oscillation. We discuss a new proof of this result in a discrete setting, and extend part of the equivalence above to the higher parameter setting.Comment: To appear in Hokkaido Math J. This is the final version of the paper. Several typos correcte

On the Hilbert transform and lacunary directions in the plane

Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert transform replaced by the one dimensional maximal function was proved by Nagel, Stein and Wainger in 1978.Comment: 11 pages, 1 fiure. This is version to appear in Illinois J. Mat

The Linear Bound in A_2 for Calder\'on-Zygmund Operators: A Survey

For an L ^2-bounded Calderon-Zygmund Operator T, and a weight w \in A_2, the norm of T on L ^2 (w) is dominated by A_2 characteristic of the weight. The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973, has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A_2 character of the weight can be exactly once in the proof. Accordingly, a large part of the proof uses two-weight techniques, is based on novel decomposition methods for operators and weights, and yields new insights into the Calder\'on-Zygmund theory. We survey the proof of this Theorem in this paper.Comment: 19 pages. Submitted to the proceedings of the Jozef Marcinkiewicz Centenary Conferenc

On the Separated Bumps Conjecture for Calderon-Zygmund Operators

We study the `separated bump conjecture' of Cruz-Uribe & Perez, and Cruz-Uribe & Reznikov & Volberg. In the L^p setting, we formulate a stronger version of this conjecture, and show that under it, a two weight inequality holds for all CZOs. When p=2, this is the result of Nazarov & Reznikov & Volberg (1306.2653). Our argument is based on stopping time arguments and the extra hypothesis is used in a clear-cut and seemingly essential way. This argument could be of some help in searching for a counterexample to the conjecture.Comment: 15 pages. v4: to appear in Hokkaido Math

Sparse Bounds for Spherical Maximal Functions

We consider the averages of a function $f$ on $\mathbb R ^{n}$ over spheres of radius $0< r< \infty$ given by $A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $\sigma$ is the normalized rotation invariant measure on $\mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function. $$M_{{lac}} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{{full}} f = \sup_{ r>0 } A_{r} f .$$ The sparse bounds are very precise variants of the known $L^p$ bounds for these maximal functions. They are derived from known $L ^{p}$-improving estimates for the localized versions of these maximal functions, and the indices in our sparse bound are sharp. We derive novel weighted inequalities for weights in the intersection of certain Muckenhoupt and reverse H\"older classes.Comment: 20 pages, 7 figures. To appear in J D'Analyse Mat

Small Ball and Discrepancy Inequalities

This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by Dmitry Bilyk and the author. The focus of that paper is a new inequality for sums of hyperbolic Haar functions in three variables, extending a famous result of J Beck from 1987. This is an improvement on what is known as the Small Ball Conjecture. In this paper, that result is proved, in a more leisurely fashion and additional remarks. In addition, background material is gathered together, including a complete proof of the necessary Harmonic Analysis; a summary of known results on the Small Ball inequality; Irregularities of Distribution; the relationship with conjectures in Approximation Theory and Probability Theory.Comment: 73 page

Issues related to Rubio de Francia's Littlewood--Paley Inequality: A Survey

Rubio de Francia's Littlewood Paley inequality is an extension of the classical Littlewood Paley inequality to one that holds for a decomposition of frequency space into arbitrary disjoint intervals. We survey this inequality, its higher dimensional analogs, and the implications for multipliers.Comment: 43 pages, 43 references. Additional material and references added. A final polishing of the manuscript. To appear in NYJ Mat

Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

A conjecture of Nazarov--Treil--Volberg on the two weight inequality for the Hilbert transform is verified. Given two non-negative Borel measures u and w on the real line, the Hilbert transform $H_u$ maps $L^2(u)$ to $L^2(w)$ if and only if the pair of measures of satisfy a Poisson $A_2$ condition, and dual collections of testing conditions, uniformly over all intervals. This strengthens a prior characterization of Lacey-Sawyer-Shen-Uriate-Tuero arxiv:1201.4319. The latter paper includes a `Global to Local' reduction. This article solves the Local problem.Comment: Final Version, to appear in Duk

An A_p --A_infty inequality for the Hilbert Transform

Continuing a theme of Lerner and Hytonen-Perez, we establish an L^p(w) inequality for a Haar shift operator of bounded complexity, that quantifies the contribution of the A_infty characteristic of the weight to the L^p norm. Here, 1<p<\infty. The Hytonen-Perez inequality is only for p=2, and we improve an inequality of the author and 6 other collaborators. As a corollary, the same inequality holds for all Calderon-Zygmund operators in the convex hull of Haar shifts of a bounded complexity, of which the canonical example is the Hilbert transform. We conjecture that the same inequality holds for all Calderon-Zygmund operators.Comment: 15 pages. (v2) Proof at end of section 4 is reworded. (v3) final version, accepted to Houston J Mat