arXiv subjects

Michael T Lacey

Publications and source records attributed to Michael T Lacey.

26 records · Page 2Linked to original sources

arXiv2006

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.

math.CA↗

arXiv2006

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 $$ R_αf(x)=\int f(x-y)\frac{dy}{\abs y ^α},\qquad 0<α<1. $$ They map $L^p\mapsto L^q$, for $1-α+\frac1q=\frac1p$, a fact we shall take for granted in this paper. A Theorem of Chanillo \cite{MR84j:42027} states that one has the equivalence $$ \norm [ M_b, R_α].p\to q.\simeq \norm b.\operatorname{BMO}. $$ 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.

math.CA↗

arXiv2005

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.

math.CA↗

arXiv2005

Paraproducts in One and Several Parameters

For multiparameter bilinear paraproduct operators $B$ we prove the estimate $$ B: L^p X L^q --> L^r, 1<p,q\le{}\infty. $$ Here, $1/p+1/q=1/r$ and special attention is paid to the case of $0<r<1$. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele. We offer a different proof of these inequalities.

math.CA↗

arXiv2004

Maximal Theorems for the Directional Hilbert Transform on the Plane

For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $ζ$ be a Schwartz function with frequency support in the annulus $1\le| ξ|\le2$. We prove that the maximal operator $$ \sup_{\abs v=1}\abs{H_vζ* f} $$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series, especially the recent proof given by Lacey and Thiele.

math.CA↗

arXiv2004

Remarks on Product VMO

Well known results related to the compactness of Hankel operators of one complex variable are extended to little Hankel operators of two complex variables. Critical to these considerations is the result of Ferguson and Lacey characterizing the boundedness of the little Hankel operators in terms of the product BMO of S.-Y. Chang and R. Fefferman.

math.CA↗

arXiv2004

An Estimate of the Maximal Operators Associated with Generalized Lacunary Sets

Let $Ω$ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $Ω$ We are interested in extensions of lacunary sets of directions, to collections we call $N$--lacunary, for integers $N$. We proceed by induction. Say that $Ω$ is 1--lacunary iff $Ω$ is an ordinary lacunary set of vectors. Every $N+1$--lacunary set can be obtained from some $N$--lacunary $Ω_N$ adding some points to $Ω_N$. Between each two neighbor points $a,b\inΩ_N$ we can add a 1--lacunary sequence (finite or infinite). We show that for all $N$ lacunary sets $Ω$, $$ \|M_Ωf(x)\|_2\lesssim{}N \|f\|_2. $$ Observe that every set $Ω$ of $N$ points is $(C\log N)$--lacunary. We then obtain a Theorem of N. Katz \cite{Katz2}. Both the current inequality, and Katz' result are consequence of a general result of Alfonseca, Soria, and Vargas \cites{ASV2}. We offer the current proof as a succinct, self--contained approach to this inequality.

math.CA↗

arXiv2004

Rubio de Francia Littlewood Paley Inequalities and Directional Maximal Functions

In $R^d$, define a maximal function in the directions $v\in \directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$ For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of $f$ to a region $\zw$. We are especially interested taking \zw to be a sector of $R^d$ with base points at the origin. A sector is a product of the interval $(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of $\directions$. Consider a collection $\zW$ of pairwise disjoint sectors $\zw$ as above. Assume that $M^\directions $ maps $L^p$ into $L^p$, for some $1<p<\zI $. Then we have the following Littlewood--Paley inequality $$ \NORm \Bigl[\sum_{\zw\in\zW}\abs{S_\zw f}^2\Bigr]^{1/2}.q.\lesssim{}\norm f.q., \qquad 2\le q<2 \frac p{p-1}. $$ The one dimensional analogue of this inequality is due to Rubio de Francia. The conclusion when the set of vectors is a fixed basis is known, is due to Journé. Our method of proof relies on a phase plane analysis. We introduce a notion of Carleson measures adapted to $\directions$, and demonstrate a John Nirenberg inequality for these measures. The John Nirenberg inequality, and an obvious $L^2$ estimate will prove the Theorem.

math.CA↗