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Michael Lacey

Publications and source records attributed to Michael Lacey.

34 records · Page 2Linked to original sources
arXiv2010

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.

math.CA↗
arXiv2008

Multiparameter Riesz Commutators

It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert transforms. The method of proof requires the real-variable methods throughout, which is new in the multi-parameter context.

math.CA↗
arXiv2008

Composition of Haar Paraproducts: The Random Case

When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variants of this question, finding non-classical characterizations, for dyadic paraproducts.

math.CA↗
arXiv2008

On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, ε}f(x) := \text{p.v.}\int_{-ε}^ εf(x-yv(x)) \frac{dy}y where $ ε$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ ε$ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.

math.CA↗
arXiv2007

On the Small Ball Inequality in All Dimensions

Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum _{|R|=2 ^{-n}} α(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^η for some positive η, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.

math.CA↗
arXiv2007

Breaking the duality in the return times theorem

We prove Bourgain's Return Times Theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages. As an immediate corollary we obtain a Wiener-Wintner type of result for the ergodic Hilbert series.

math.DS↗
arXiv2006

On a Lipschitz Variant of the Kakeya Maximal Function

In a prior work [Hilbert transform along smooth families of lines math.CA/0310345] the authors introduced a variant of the Kakeya maximal function associated with Lipschitz maps from the plane into the unit circle. In this paper, we improve the known estimates for this maximal operator--and raise the conjecture that the bounds established are optimal.

math.CA↗
arXiv2006

Lectures on Nehari's Theorem on the Polydisk

We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144) and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little" Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is the product BMO space identified by Chang and Fefferman. This article begins with the classical Nehari theorem, and presents the necessary background for the proof of the extension above. The proof of the extension is an induction on parameters, with a bootstrapping argument. Some of the more technical details of the earlier proofs are now seen as consequences of a paraproduct theory.

math.CA↗
arXiv2006

Wiener-Wintner for Hilbert Transform

We prove the following extension of the Wiener--Wintner Theorem in Ergodic Theor and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows $ (X,μ, T_t)$ and $ f\in L^p (X,μ)$, there is a set $X_f\subset X $ of probability one, so that for all $x\in X_f$ we have \begin{equation*} \lim _{s\downarrow0} \int _{s<\abs t<1/s} \operatorname e ^{i θt} f(\operatorname T_tx)\; \frac{dt}t \qquad \text{exists for all $θ$.} \end{equation*} The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson's theorem.

math.CA↗
arXiv2005

Carleson's Theorem: Proof, Complements, Variations

Carleson's Theorem asserts the pointwise convergence of Fourier series of square integrable functions. We give a complete proof, following joint work of the author and C. Thiele. Over 20 exercises are also detailed. We also discuss the derviation of the theorem on L^p spaces. And discuss a number of complements and extensions of the theorem and proof of Carleson's theorem.

math.CA↗
arXiv2005

Variations on the Theme of Journe's Lemma

Journe's Lemma is a critical component of many questions related to the product ${BMO}$ theory of S.-Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, some known, some implicit in the literature, and some new.

math.CA↗
arXiv2002

A characterization of product BMO by commutators

Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm of b. Here, M_b denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H^1 to the same theorem on H^1 of a product domain. The product setting is more delicate because of the presence of a two parameter family of dilations. The method of proof depends upon (a) A dyadic decomposition of product BMO by wavelets (b) a prior estimate of Ferguson and Sadosky involving rectangular BMO (c) and a careful control of certain measures related to those of Carleson.

math.CA↗
arXiv2001

Carleson's theorem with quadratic phase

Carleson's theorem on the pointwise convergence of Fourier series provides bounds for a maximal operator, with the maximum taken over all choices of linear functions of a phase argument. We extend this to all quadratic choices of phase functions. Specifically, we show that the maximal operator below maps $L^p$ into itself for $1<p<\infty$. $$ \sup_a \sup_b |\int e^{i(ay^2+by)}f(x-y)dy/y| $$

math.CA↗
arXiv1999

On Calderón's conjecture

This paper is a successor of \cite{laceyt}. In that paper we considered bilinear operators of the form H_alpha(f_1,f_2)(x) = p.v. \int f_1(x-t) f_2(x + alpha t)/t dt, which are originally defined for f_1, f_2 in the Schwartz class S(R). The natural question is whether estimates of the form H_alpha(f_1,f_2)|_p <= C_{alpha,p_1,p_2} |f_1|_{p_1} |f_2|_{p_2} with constants C_{alpha,p_1,p_2} depending only on alpha,p_1,p_2 and p = p_1p_2/(p_1+p_2) hold. The purpose of the current paper is to extend the range of exponents p_1 and p_2 for which the estimate is known. In particular, the case p_1=2, p_2=\infty is solved to the affirmative. This was originally considered to be the most natural case and is known as Calderón's conjecture.

math.CA↗