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Michael Lacey

Publications and source records attributed to Michael Lacey.

At least 19 recordsLinked to original sources
arXiv2026

Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

math.CA↗
arXiv2026

The Wiener Wintner Theorem Along the Primes

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, \nu),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $\nu(X_f) = 1,$ so that for all $\omega \in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 \pi i p_n \theta} f(T^{p_n} \omega) \] converges for all $\theta \in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.

math.DS↗
arXiv2024

Schatten classes and commutators in the two weight setting, II. Riesz transforms

We characterize the Schatten class $S^p$ of the commutator of Riesz transforms $[b,R_j]$ in $\mathbb R^n$ ($j=1,\ldots, n$) in the two weight setting for $n< p<\infty$, by introducing the condition that the symbol $b$ being in Besov spaces associated with the given two weights. At the critical index $p=n$, the commutator $[b,R_j]$ belongs to Schatten class $S^{n}$ if and only if $b$ is a constant, and to the weak Schatten class $S^{n,\infty}$ if and only if $b$ is in an oscillation sequence space associated with the given two weights. As a direct application, we have the Schatten class estimate for A. Connes' quantised derivative in the two weight setting.

math.CA↗
arXiv2024

Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators

Let $\Delta_\lambda$ be the Bessel operator on the upper half space $\mathbb{R}_+^{n+1}$ with $n\geq 0$ and $\lambda>0$, and $R_{\lambda,j}$ be the $j-$th Bessel Riesz transform, $j=1,\ldots,n+1$. We demonstrate that the Schatten--Lorentz norm ($S^{p,q}$, $1<p<\infty$, $1\leq q\leq \infty$) of the commutator $[b,R_{\lambda,j}]$ can be characterized in terms of the oscillation space norm of the symbol $b$. In particular, for the case $p=q$, the Schatten norm of $[b,R_{\lambda,j}]$ can be further characterized in terms of the Besov norm of the symbol. Moreover, the critical index is also studied, which is $p=n+1$, the lower dimension of the Bessel measure (but not the upper dimension). Our approach relies on martingale and dyadic analysis, which enables us to bypass the use of Fourier analysis effectively.

math.FA↗
arXiv2023

Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus \omega ) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in \omega $, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.

math.NT↗
arXiv2022

Compactness of the Bloom sparse operators and applications

We establish the characterization of compactness for the sparse operator (associated with symbol in weighted VMO space) in the two weight setting on the spaces of homogeneous type in the sense of Coifman and Weiss. As a direct application we obtain the compactness characterization for the maximal commutators with respect to the weighted VMO functions and the commutator of Calder\'on-Zygmund operators on the homogeneous spaces. Furthermore, our approach can be applied to compactness characterization for operators in the multilinear Bloom setting.

math.CA↗
arXiv2022

Schatten classes and commutator in the two weight setting, I. Hilbert transform

We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

math.FA↗
arXiv2021

Schatten classes and commutators of Riesz transform on Heisenberg group and applications

We study commutators with the Riesz transforms on the Heisenberg group. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson--Wolff and Rochberg--Semmes. The method of proof extends the earlier methods, allowing us to address not just the Riesz transforms, but also the Cauchy--Szego projection and second order Riesz transforms on the Heisenberg group, among other settings.

math.FA↗
arXiv2020

Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón--Zygmund operator $T$, and a pair of weights $ σ, ω\in A_p$, the commutator $ [T, b]$ is compact from $ L ^{p} (σ) \to L ^{p} (ω)$ if and only if $ b \in VMO _{ν}$, where $ ν= (σ/ ω) ^{1/p}$. This extends the work of the first author, Holmes and Wick. The weighted $VMO$ spaces are different from the classical $ VMO$ space. In dimension $ d =1$, compactly supported and smooth functions are dense in $ VMO _{ν}$, but this need not hold in dimensions $ d \geq 2$. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.

math.CA↗
arXiv2020

Averages Along the Primes: Improving and Sparse Bounds

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds \begin{equation*} N ^{-1/p' }\lVert \mathcal{A}_N f\rVert_{\ell^{p'}} \leq C_p N ^{- 1/p} \lVert f\rVert_{\ell^p}. \end{equation*} The maximal function $ \mathcal{A}^{\ast} f =\sup_{N} \lvert \mathcal{A}_N f \rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ \mathcal{A}^{\ast} $ is bounded on $ \ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ \mathcal{A}^{\ast} $ were known.

math.CA↗
arXiv2020

Improving estimates for discrete polynomial averages

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.

math.CA↗
arXiv2019

Commutators of Cauchy--Szegő type integrals for domains in $\mathbb C^n$ with minimal smoothness

In this paper we study the commutator of Cauchy type integrals $\EuScript C$ on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ as in the recent result of Lanzani--Stein. We point out that in this setting the Cauchy type integrals $\EuScript C$ is the sum of the essential part $\EuScript C^\sharp$ which is a Calderón--Zygmund operator and a remainder $\EuScript R$ which is no longer a Calderón--Zygmund operator. We show that the commutator $[b, \EuScript C]$ is bounded on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the BMO space on $bD$. Moreover, the commutator $[b, \EuScript C]$ is compact on $L^p(bD)$ ($1<p<\infty$) if {\color{black}and only if}\ $b$ is in the VMO space on $bD$. Our method can also be applied to the commutator of Cauchy--Leray integral in a bounded, strongly $\mathbb C$-linearly convex domain $D$ in $\mathbb C^n$ with the boundary $bD$ satisfying the minimum regularity $C^{1,1}$. Such a Cauchy--Leray integral is a Calderón--Zygmund operator as proved in the recent result of Lanzani--Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy--Szeg\H o operator on a bounded strongly pseudoconvex domain $D$ in $\mathbb C^n$ with smooth boundary (first established by Krantz--Li).

math.CV↗
arXiv2019

Endpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type

We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{\frac12}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.

math.CA↗
arXiv2017

On Convergence of Oscillatory Ergodic Hilbert Transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, $(X,Σ,μ,τ)$, with $τ$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform \[ \sum_{n=1}^\infty \frac{e^{2πi p(n)}}{n} τ^n f(x) \] converges $μ$-a.e. for each $f \in L^r(X), \ 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, \[ \sum_{n=1}^\infty \frac{X_n}{n} τ^n f(x), \] where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.

math.CA↗
arXiv2016

A Discrete Quadratic Carleson Theorem on $ \ell ^2 $ with a Restricted Supremum

Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$ functions \[ \mathcal{C}_Λ f( n ) := \sup_{ λ\in Λ} \left| \sum_{m \neq 0} f(n-m) \frac{e^{2 πiλm^2}} {m} \right| \] where $Λ\subset [0,1]$. We give sufficient conditions on $Λ$, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E. M. Stein's integral result, that the maximal operator below is bounded on $L^2(\mathbb R)$. \[ \mathcal{C}_2 f(x):= \sup_{λ\in \mathbb R} \left| \int f(x-y) \frac{e^{2πi λy^2}}{y} \ dy \right|.\] The proof of our result relies heavily on Bourgain's work on arithmetic ergodic theorems, with novel complexity arising from the oscillatory nature of the question at hand, and difficulties arising from the the parameter $λ$ above.

math.CA↗
arXiv2012

Weighted bounds for variational Fourier series

For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.

math.CA↗
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↗