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Michael T Lacey

Publications and source records attributed to Michael T Lacey.

At least 19 recordsLinked to original sources
arXiv2021

A Polynomial Roth Theorem for Corners in Finite Fields

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $ϕ_1$ and $ϕ_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with $ \lvert A\rvert > p ^{2 - \frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + ϕ_1 (y), x_2), (x_1, x_2 + ϕ_2 (y))$. The study of these questions on $ \mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz.

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arXiv2020

Averages along the Square Integers: $\ell^p$ improving and Sparse Inequalities

Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$: \begin{equation*} N ^{-2/p'} \lVert A_N f \rVert _{ p'} \lesssim N ^{-2/p} \lVert f\rVert _{\ell ^{p}}, \end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \sup _{N\geq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.

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arXiv2014

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.

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arXiv2014

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.

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arXiv2014

Two weight norm inequalities for the $g$ function

Given two weights $σ, w$ on $\mathbb R ^{n}$, the classical $g$-function satisfies the norm inequality $\lVert g (fσ)\rVert_{L ^2 (w)} \lesssim \lVert f\rVert_{L ^2 (σ)}$ if and only if the two weight Muckenhoupt $A_2$ condition holds, and a family of testing conditions holds, namely \begin{equation*} \iint_{Q (I)} (\nabla P_t (σ\mathbf 1_I)(x, t))^2 \; dw \, t dt \lesssim σ(I) \end{equation*} uniformly over all cubes $I \subset \mathbb R ^{n}$, and $Q (I)$ is the Carleson box over $I$. A corresponding characterization for the intrinsic square function of Wilson also holds.

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arXiv2013

Dichotomy Results for the L1 Norm of the Discrepancy Function

It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been established by Halasz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L^1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.

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arXiv2013

Estimates of the Discrepancy Function in Exponential Orlicz Spaces

We prove that in all dimensions n at least 3, for every integer N there exists a distribution of points of cardinality $ N$, for which the associated discrepancy function D_N satisfies the estimate an estimate, of sharp growth rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently been proved by M.~Skriganov, using random digit shifts of binary digital nets, building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our approach, developed independently, complements that of Skriganov.

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arXiv2012

Weighted Weak Type Estimates for Square Functions

We consider the weak-type inequality for Littlewood-Paley square functions on A_p weighted Lebesgue spaces. Of interest is the sharp in the A_p characteristic estimate. The case of 1<p<2 is subcritical, and the sharp power of 1/p is established. In the critical case of p=2, we miss the critical exponent 1/2 by a logarithm of the A_p characteristic. These estimates improve on known estimates for 1<p<3.

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arXiv2012

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.

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arXiv2012

On the Local $ Tb$ Theorem under Minimal Integrability

We prove a version of the local Tb Theorem assuming that the accretive functions b_Q and T b_Q are locally L ^{p} integrable, for any 1< p < \infty . This improves a recent result of Hytonen-Nazarov. The proof strategy relies upon the their strategy, with additional techniques concerning twisted martingale differences and the use of random dyadic grids in the local Tb setting.

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arXiv2011

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.

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arXiv2010

The Linear Bound in A_2 for Calderón-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 Calderón-Zygmund theory. We survey the proof of this Theorem in this paper.

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arXiv2007

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}.

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arXiv2007

On the Signed Small Ball Inequality

This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions.' In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved.

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arXiv2007

On the Small Ball Inequality in Three Dimensions

We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$. We show that there is a postive $η$ so that for all integers $n$, and coefficients $ α(R)$ we have 2 ^{-n} \sum_{\abs{R}=2 ^{-n}} \abs{α(R)} {}\lesssim{} n ^{1 - η} \NOrm \sum_{\abs{R}=2 ^{-n}} α(R) h_R >.\infty . This is an improvement over the `trivial' estimate by an amount of $n ^{- η}$, and the optimal value of $η$ (which we do not prove) would be $ η=\frac12$. There is a corresponding lower bound on the $L ^{\infty}$ norm of the Discrepancy function of an arbitary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension 3, is that of J{ó}zsef Beck \cite{MR1032337}, in which the improvement over the trivial estimate was logarithmic in $n$. We find several simplifications and extensions of Beck's argument to prove the result above.

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arXiv2007

Hankel Operators in Several Complex Variables and Product $BMO\zProd$

$H^2\zProd$ denotes the Hardy space of square integrable functions analytic in each variable separately. Let $P^{\ominus}$ be the natural projection of $L^2\zProd$ onto $\z8{H^2\zProd}$. A Hankel operator with symbol $b$ is the linear operator from $H^2\zProd$ to $\z8{H^2\zProd}$ given by $H_b \zvf=P^{\ominus}\bar b \zvf$. We show that \md0 \norm H_b ..\simeq \norm P^{\oplus}b.BMO\zProd., \emd where the right hand norm is S.-Y. Chang and R. Fefferman product $BMO$. This fact has well known equivalences in terms of commutators and the weak factorization of $H^1\zProd$. In the case of two complex variables, this is due to Ferguson and Lacey \cite{MR1961195}. While the current proof is inductive, and one can take the one complex variable case as the basis step, it is heavily influenced by the methods of Ferguson and Lacey. The induction is carried out with a particular form of a lemma due to Journé \cite{MR87g:42028}, which occurs implicitly in the work of J. Pipher \cite{MR88a:42019}.

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arXiv2007

On an Argument of Shkredov on Two-Dimensional Corners

Let $\mathbb F_2^n$ be the finite field of cardinality $2 ^{n}$. For all large $n$, any subset $A\subset \mathbb F_2^n\times \mathbb F_2 ^n$ of cardinality \begin{equation*} \abs{A} \gtrsim 4^n \log\log n (\log n) ^{-1} \end{equation*} must contain three points $ \{(x,y) ,(x+d,y) ,(x,y+d)\}$ for $x,y,d\in \mathbb F_2^n$ and $d\neq0$. Our argument is an elaboration of an argument of Shkredov \cite {math.NT/0405406}, building upon the finite field analog of Ben Green \cite {math.NT/0409420}. The interest in our result is in the exponent on $ \log n$, which is larger than has been obtained previously.

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