arXiv subjects

Michael T. Lacey

Publications and source records attributed to Michael T. Lacey.

At least 37 records · Page 2Linked to original sources

arXiv2018

Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

For $ 0< λ< \frac{1}2$, let $ B_{λ}$ be the Bochner-Riesz multiplier of index $ λ$ on the plane. Associated to this multiplier is the critical index $1 < p_λ= \frac{4} {3+2 λ} < \frac{4}3$. We prove a sparse bound for $ B_{λ}$ with indices $ (p_λ, q)$, where $ p_λ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_λ}$ boundedness of $ B_{λ}$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 \cap RH_{ρ}$, where $ ρ> \frac4 {4 - 3 p_{λ}}$.

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arXiv2017

Sparse Bounds for Bochner-Riesz Multipliers

The Bochner-Riesz multipliers $ B_{δ}$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< δ< \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ δ$ increases to the critical value, $ δ=\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.

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arXiv2017

Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms

For each $ d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $ (1, r )$ sparse bound, for all $ r>1$ $$ H _{ \ast } f (x) = \sup _{ε} \Bigl\lvert \int_{|y| > ε} f (x-y) \frac { e ^{2 πi y ^d }} y\; dy \Bigr\rvert. $$ This quickly implies weak-type inequalities for the maximal truncations, which hold for $A_1$ weights, but are new even in the case of Lebesgue measure. The unweighted weak-type estimate \emph{without maximal truncations} but with arbitrary polynomials, is due to Chanillo and Christ (1987).

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arXiv2017

Sparse Bounds for Oscillatory and Random Singular Integrals

Let $ T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy $, where $ K (y)$ is a smooth Calderón-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g \rangle$. This in turn easily implies $ A_p $ inequalities. The method of proof is applied in a random discrete setting, yielding the first weighted inequalities for operators defined on sparse sets of integers.

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arXiv2016

The Sparse T1 Theorem

We impose standard $ T1 $-type assumptions on a Calderón-Zygmund operator $ T $, and deduce that for bounded compactly supported functions $ f, g $ there is a sparse bilinear form $ Λ$ so that $$ \lvert \langle T f, g \rangle\rvert \lesssim Λ(f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $ L ^{p}$ space.

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arXiv2016

One Bit Sensing, Discrepancy, and Stolarsky Principle

A sign-linear one bit map from the $ d$-dimensional sphere $ \mathbb S ^{d}$ to the $ n$-dimensional Hamming cube $ H^n= \{ -1, +1\} ^{n}$ is given by $$ x \to \{ \mbox{sign} (x \cdot z_j) \;:\; 1\leq j \leq n\} $$ where $ \{z_j\} \subset \mathbb S ^{d}$. For $ 0 < δ< 1$, we estimate $ N (d, δ)$, the smallest integer $ n$ so that there is a sign-linear map which has the $ δ$-restricted isometric property, where we impose normalized geodesic distance on $ \mathbb S ^{d}$, and Hamming metric on $ H^n$. Up to a polylogarithmic factor, $ N (d, δ) \approx δ^{-2 + \frac2{d+1}}$, which has a dimensional correction in the power of $ δ$. This is a question that arises from the one bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the $L^2$ average of the embedding error is equivalent to minimizing the discrete energy $\sum_{i,j} \big( \frac12 - d(z_i,z_j) \big)^2$, where $d$ is the normalized geodesic distance.

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arXiv2016

Sparse Bounds for Random Discrete Carleson Theorems

We study discrete random variants of the Carleson maximal operator. Intriguingly, these questions remain subtle and difficult, even in this setting. Let $\{X_m\}$ be an independent sequence of $\{0,1\}$ random variables with expectations \[ \mathbb E X_m = σ_m = m^{-α}, \ 0 < α< 1/2, \] and $ S_m = \sum_{k=1} ^{m} X_k$. Then the maximal operator below almost surely is bounded from $ \ell ^{p}$ to $ \ell ^{p}$, provided the Minkowski dimension of $ Λ\subset [-1/2, 1/2]$ is strictly less than $ 1- α$. \[ \sup_{λ\in Λ} \Bigl| \sum_{m\neq 0} X_{\lvert m\rvert } \frac{e( λm )}{ {\rm sgn} (m)S_{ |m| }} f(x- m) \Bigr|. \] This operator also satisfies a sparse type bound. The form of the sparse bound immediately implies weighted estimates in all $ \ell ^{2}$, which are novel in this setting. Variants and extensions are also considered.

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arXiv2016

Weak Type Bound for Oscillatory Singular Integrals

Let $ T _{P} f (x) = \int e ^{i P (y)} K (y) f (x-y) \, dy $, where $ K (y)$ is a smooth Calderón-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. The maximal truncations of $ T_P$ satisfy the weak $ L ^{1}$ inequality, our proof simplifying and extending the argument of Chanillo and Christ for the weak type bound for $ T_P$.

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arXiv2016

Two Weight Inequalities for Riesz Transforms: Uniformly Full Dimension Weights

Fix an integer $ n$ and number $d$, $ 0< d\neq n-1 \leq n$, and two weights $ w$ and $ σ$ on $ \mathbb R ^{n}$. We two extra conditions (1) no common point masses and (2) the two weights separately are not concentrated on a set of codimension one, uniformly over locations and scales. (This condition holds for doubling weights.) Then, we characterize the two weight inequality for the $ d$-dimensional Riesz transform on $ \mathbb R ^{n}$, \begin{equation*} \sup_{0< a < b < \infty}\left\lVert \int_{a < \lvert x-y\rvert < b} f (y) \frac {x-y} {\lvert x-y\rvert ^{d+1}} \; σ(dy) \right\rVert_{L ^{2} (\mathbb{R}^n;w)} \le \mathscr N \lVert f\rVert_{L ^2 (\mathbb{R}^n;σ)} \end{equation*} in terms of these two conditions, and their duals: For finite constants $ \mathscr A_2$ and $ \mathscr T$, uniformly over all cubes $ Q\subset \mathbb R ^{n}$ \begin{gather*} \frac {w (Q)} {\lvert Q\rvert ^{d/n}} \int_{\mathbb R ^{n}} \frac {\lvert Q\rvert ^{d/n}} {\lvert Q\rvert ^{2d/n} +{dist}(x, Q) ^{2d/n}} \; σ(dx) \leq \mathscr A_2 \\ \int_{Q} \lvert \mathsf R_σ \mathbf 1_{Q} (x)\rvert ^2 \; w(dx) \le \mathscr T ^2 σ(Q), \end{gather*} where $ \mathsf R_σ$ denotes any of the truncations of the Riesz transform as above, the dual conditions are obtained by interchanging the roles of the two weights. Examples show that a key step of the proof fails in absence of the extra geometric condition imposed on the weights.

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arXiv2016

On $A_p$-$A_\infty$ type estimates for square functions

We prove strong-type $A_p$-$A_\infty$ estimate for square functions, improving on the $ A_p$ bound due to Lerner. Entropy bounds, in the recent innovation of Treil-Volberg, are then proved. The techniques of proof include parallel stopping cubes, pigeon-hole arguments, and the approach to entropy bounds of Lacey--Spencer.

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arXiv2016

Commutators in the Two-Weight Setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $μ,λ\in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;μ) \to L^p(\mathbb{R}^n;λ)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $μ$ and $λ$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $λ$ and $μ$ being Lebesgue measure, and Bloom in the case of dimension one.

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arXiv2015

Random Tessellations, Restricted Isometric Embeddings, and One Bit Sensing

We obtain mproved bounds for one bit sensing. For instance, let $ K_s$ denote the set of $ s$-sparse unit vectors in the sphere $ \mathbb S ^{n}$ in dimension $ n+1$ with sparsity parameter $ 0 < s < n+1$ and assume that $ 0 < δ< 1$. We show that for $ m \gtrsim δ^{-2} s \log \frac ns$, the one-bit map $$ x \mapsto \bigl[ {sgn} \langle x,g_j \rangle \bigr] _{j=1} ^{m}, $$ where $ g_j$ are iid gaussian vectors on $ \mathbb R ^{n+1}$, with high probability has $ δ$-RIP from $ K_s$ into the $ m$-dimensional Hamming cube. These bounds match the bounds for the {linear} $ δ$-RIP given by $ x \mapsto \frac 1m[\langle x,g_j \rangle ] _{j=1} ^{m} $, from the sparse vectors in $ \mathbb R ^{n}$ into $ \ell ^{1}$. In other words, the one bit and linear RIPs are equally effective. There are corresponding improvements for other one-bit properties, such as the sign-product RIP property.

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arXiv2015

Bloom's Inequality: Commutators in a Two-Weight Setting

In 1985, Bloom characterized the boundedness of the commutator $[b,H]$ as a map between a pair of weighted $L^{p}$ spaces, where both weights are in $A_p$. The characterization is in terms of a novel $BMO$ condition. We give a 'modern' proof of this result, in the case of $p=2$. In a subsequent paper, this argument will be used to generalize Bloom's result to all Calderón-Zygmund operators and dimensions.

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arXiv2015

An elementary proof of the $A_2$ Bound

A martingale transform $ T$, applied to an integrable locally supported function $ f$, is pointwise dominated by a positive sparse operator applied to $ \lvert f\rvert $, the choice of sparse operator being a function of $ T$ and $ f$. As a corollary, one derives the sharp $ A_p$ bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-$L^1$ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the $ A_2$ bounds in that setting.

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arXiv2015

The Two Weight Inequality for the Hilbert Transform: A Primer

Recent work of Lacey-Sawyer-Shen-Uriarte-Tuero and Lacey have established a conjecture of Nazarov-Treil-Volberg, giving a real-variable characterization of the two weight inequality for the Hilbert transform, provided the pair of weights do not share a common point mass. In this paper, the characterization is proved, collecting details from across several papers, compactness is characterized, all relevant estimates are proved, counterexamples are details, and areas of application are indicated.

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arXiv2015

On Entropy Bumps for Calderón-Zygmund Operators

We study two weight inequalities in the recent innovative language of `entropy' due to Treil-Volberg. The inequalities are extended to $ L ^{p}$, for $ 1< p \neq 2 < \infty $, with new short proofs. A result proved is as follows. Let $ \varepsilon $ be a monotonic increasing function on $ (1, \infty)$ which satisfy $ \int _{1} ^{\infty} \frac {dt} {\varepsilon (t) t} = 1$. Let $ σ$ and $ w$ be two weights on $ \mathbb R ^{d}$. If this supremum is finite, for a choice of $ 1< p < \infty $, $$ \sup _{Q} \biggl[ \frac {σ(Q)} {\lvert Q\rvert} \biggr]^{p-1} \frac {\int _{Q} M (σχ_{Q})} {σ(Q)} \cdot \frac {w (Q)} {\lvert Q\rvert}\biggl[ \frac {\int _{Q} M (w χ_{Q})} {w (Q)}\biggr]^{p-1} < \infty, $$ then any Calderón-Zygmund operator $ T$ satisfies the bound $ \lVert T _σ f \rVert _{L ^{p} (w)} \lesssim \lVert f\rVert _{L ^{p} (σ)} $.

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arXiv2015

Borderline Weak Type Estimates for Singular Integrals and Square Functions

For any Calderón-Zygmund operator $ T$, any weight $ w$, and $ α>1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{α} } w )$ into weak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, Pérez, and Hytönen-Pérez, on the $ L (\log L) ^{ε}$ scale. Also, for square functions $ S f$, and weights $ w \in A_p$, the norm of $ S$ from $ L ^p (w)$ to weak-$L^p (w)$, $ 2\leq p < \infty $, is bounded by $ [w] _{A_p}^{1/2} (1+\log [w] _{A_ \infty }) ^{1/2} $, which is a sharp estimate.

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arXiv2014

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

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to weak-L^2 inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions.

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