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Eric T. Sawyer

Publications and source records attributed to Eric T. Sawyer.

At least 19 recordsLinked to original sources
arXiv2026

A discussion of three arguments related to Fefferman's Fourier extension theorem in the plane

The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. However, each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer. On the other hand, our third argument avoids both of these obstacles by using smooth Alpert projections with wave packets, discrete bilinear characterizations, and the discrete Fourier transform of the coefficient sequences associated with the projections.

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arXiv2026

A probabilistic analogue of the Fourier extension conjecture

We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and interpolation of L^2 and L^4 estimates. The correct L^4 bounds for resonant forms require an expectation over Alpert multipliers.

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arXiv2025

A reprise of the NTV conjecture for the Hilbert transform

We give a slightly different proof of the NTV conjecture for the Hilbert transform that was proved by T. Hytönen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero, building on previous work of F. Nazarov, S. Treil and A. Volberg. After modifying the decomposition of the main bilinear form, we give a new proof of control of functional energy that is based on the potential Theorem 1 of [Saw3], rather than the Poisson Theorem 2 that is used in all other proofs in the literature. This approach was pioneered in the first version of Sawyer and Wick [SaWi] on the ArXiv. Then we alter the bottom-up corona construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains as in the fundamental work of Lacey.

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arXiv2024

The Hytönen-Vuorinen L^{p} conjecture for the Hilbert transform, with an extended energy side condition, when (4/3)<p<4 and the measures share no point masses

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hytönen and E. Vuorinen from 2018 on two weight testing theorems for the Hilbert transform on weighted L^{p} spaces, but with extended energy side conditions. Namely, assuming the extended energy conditions, the two weight norm inequality holds (1) if and only if the global quadratic interval testing conditions hold, (2) if and only if the local quadratic interval testing, the quadratic Muckenhoupt, and the quadratic weak boundedness conditions all hold. We also give a slight improvement of the second conjecture in this setting by replacing the quadratic Muckenhoupt conditions with two smaller conditions.

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arXiv2024

Haar basis testing

We show that for two doubling measures $σ$ and $ω$ on $\mathbb{R}^{n}$ and any fixed dyadic grid $\mathcal{D}$ in $\mathbb{R}^{n}$, \[ \mathfrak{N}_{\mathbf{R}^{λ, n}}\left( σ,ω\right) \approx\mathfrak{H}_{\mathbf{R}^{λ, n}}^{\mathcal{D},\operatorname*{glob}}\left( σ,ω\right) +\mathfrak{H}_{\mathbf{R}^{λ, n}}^{\mathcal{D},\operatorname*{glob}}\left( ω, σ\right) \ , \] where $\mathfrak{N}_{\mathbf{R}^{λ, n}} (σ, ω)$ denotes the $L^2 (σ) \to L^2 (ω)$ operator norm of the vector-Riesz transform $\mathbf{R}^{λ, n}$ of fractional order $λ\neq 1$, and \[ \mathfrak{H}_{\mathbf{R}^{λ,n}}^{\mathcal{D},\operatorname*{glob}}\left( σ,ω\right) \equiv\sup_{I\in\mathcal{D}}\left\Vert \mathbf{R}^{λ,n} h_{I}^σ\right\Vert _{L^{2}\left( ω\right) }\ , \] is the global Haar testing characteristic for $\mathbf{R}^{λ,n}$ on the grid $\mathcal{D}$, and $\left\{ h_{I}^σ\right\} _{I\in\mathcal{D}}$ is the weighted Haar orthonormal basis of $L^{2}\left( σ\right) $ arising in the work of Nazarov, Treil and Volberg. We also show this theorem extends more generally to weighted Alpert wavelets which replace the weighted Haar wavelets in the proofs of some recent two-weight $T1$ theorems. Finally, we briefly pose these questions in the context of orthonormal bases in arbitrary Hilbert spaces.

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arXiv2024

Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures

If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hytönen and Vuorinen hold. We also show that these quadratic triple testing conditions can be relaxed to quadratic local testing conditions, quadratic offset Muckenhoupt conditions, and a quadratic weak boundedness property.

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arXiv2024

Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights

We prove a T1 theorem for fractional vector Riesz transforms mapping one weighted Sobolev space to another, where the weights are doubling measures on Euclidean space. Boundedness is characterized by the classical A_2 condition and two dual testing conditions on indicators of cubes. We also show the equivalence of various weighted Sobolev norms when the measure is doubling, something that fails in general.

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arXiv2024

A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II

We consider the weak to strong type problem for two weight norm inequalities for Calderón-Zygmund operators with doubling weights. We show that if a Calderón-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T^{*} holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calderón-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T_{*} are equivalent. Thus the characterization of weak type for T_{*} in [LaSaUr1] applies to T as well.

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arXiv2022

Sum of squares I: scalar functions

This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinite regime. We give a sharp sufficient condition on a smooth nonnegative function f on n-dimensional Euclidean space so that it can be written as a finite sum of squares of C^2,delta functions. Special consideration is given to analyzing the case when f vanishes only at the origin, answering a question of Bony et al.

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arXiv2021

A T1 theorem for general Calderón-Zygmund operators with comparable doubling weights, and optimal cancellation conditions

We begin an investigation into extending the T1 theorem of David and Journé, and the corresponding cancellation conditions of Stein, to more general pairs of distinct doubling weights. For example, assuming the measures satisfy a fractional A infinity condition and are comparable in the sense of Coifman and Fefferman, we characterize the two weight norm inequality for a strongly elliptic fractional Calderón-Zygmund singular integral, in terms of the one-tailed fractional Muckenhoupt conditions, and the usual cube testing conditions. We then apply this result to give a version, in the setting of two comparable fractional A infinity weights, of Stein's characterization of cancellation conditions on a kernel K in order that there exists a bounded operator T that is associated with K. More generally we prove a T1 theorem involving a bilinear indicator/cube testing inequality in place of the weak boundedness property of David and Journeé - where we must test over all bounded functions instead of just Holder continuous functions. We use a proof strategy based on an adaptation of the `pivotal' argument of Nazarov, Treil and Volberg to the weighted Alpert wavelets of Rahm, Sawyer and Wick using a Parallel Corona decomposition of Lacey, Sawyer, Shen and Uriarte-Tuero.

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arXiv2021

Sums of squares III: hypoellipticity in the infinitely degenerate regime

This is the third in a series of papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^2,delta generalization of M. Christ's sum of squares theorem, and use a bootstrap argument with the sum of squares theorem for matrix functions in the second paper of this series, in order to prove a hypoellipticity theorem generalizing work in the infinitely degenerate regime to include nondiagonal operators and more general degeneracies.

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arXiv2021

Sums of squares II: matrix functions

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional Euclidean space, whose determinant vanishes only at the origin and such that F is comparable to its diagonal matrix, in order that F is a finite sum of squares of C^2,delta vector fields. We also consider slightly more general decompositions in which a single quasiconformal term need not be a sum of squares.

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arXiv2020

A two weight local $Tb$ theorem for $n$-dimensional fractional singular integrals

We obtain a local two weight $Tb$ theorem with an energy side condition for higher dimensional fractional Calderón-Zygmund operators. The proof follows the general outline of the proof for the corresponding one-dimensional $Tb$ theorem in [SaShUr12], but encountering a number of new challenges, including several arising from the failure in higher dimensions of T. Hytönen's one-dimensional two weight $A_{2}$ inequality [Hyt].

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