Eric T. Sawyer
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View article: Trilinear characterizations of the Fourier extension conjecture on the paraboloid in three dimensions
Trilinear characterizations of the Fourier extension conjecture on the paraboloid in three dimensions Open
We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such i…
View article: Stability of weighted norm inequalities
Stability of weighted norm inequalities Open
We show that while individual Riesz transforms are two-weight norm stable under biLipschitz change of variables on A_{\infty} weights, they are two-weight norm unstable under even rotational change of variables on doubling weights. More pr…
View article: Sums of squares III: Hypoellipticity in the infinitely degenerate regime
Sums of squares III: Hypoellipticity in the infinitely degenerate regime Open
This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^{2,\delta} generalization of M. Christ’s smooth sum of squares theorem, and then use a boot…
View article: Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights
Correction to: Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights Open
View article: A Helmholtz-type decomposition for the space of symmetric matrices
A Helmholtz-type decomposition for the space of symmetric matrices Open
In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better unde…
View article: A probabilistic analogue of the Fourier extension conjecture
A probabilistic analogue of the Fourier extension conjecture Open
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 interpol…
View article: Haar basis testing
Haar basis testing Open
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},\…
View article: The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2
The scalar $T1$ theorem for pairs of doubling measures fails for Riesz transforms when p not 2 Open
We show that for an individual Riesz transform in the setting of doubling measures, the scalar $T1$ theorem fails when $p \neq 2$: for each $ p \in (1, \infty) \setminus \{2\}$, we construct a pair of doubling measures $(σ, ω)$ on $\mathbb…
View article: The Hytönen-Vuorinen L^{p} conjecture for the Hilbert transform, with an extended energy side condition, when (4/3)
The Hytönen-Vuorinen L^{p} conjecture for the Hilbert transform, with an extended energy side condition, when (4/3) Open
In the case (4/3)
View article: The Moser method and boundedness of solutions to infinitely degenerate elliptic equations
The Moser method and boundedness of solutions to infinitely degenerate elliptic equations Open
We show that if $\mathbb{R}^{n}$ is equipped with certain non-doubling metric and an Orlicz-Sobolev inequality holds for a special family of Young functions $Φ$, then weak solutions to quasilinear infinitely degenerate elliptic divergence …
View article: A reprise of the NTV conjecture for the Hilbert transform
A reprise of the NTV conjecture for the Hilbert transform Open
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. Aft…
View article: The T1 theorem for the Hilbert transform fails when p is not 2
The T1 theorem for the Hilbert transform fails when p is not 2 Open
Given p between 1 and infinity, but not 2, we show that the T1 theorem for the Hilbert transform fails for L^{p}, despite holding for p equal to 2
View article: Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures
Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures Open
If T is a fractional vector Riesz transform, 1
View article: Stability of Weighted Norm Inequalities
Stability of Weighted Norm Inequalities Open
We show that while individual Riesz transforms are two weight norm stable under biLipschitz change of variables on $A_{\infty}$ weights, they are two weight norm unstable under even rotational change of variables on doubling weights. More …
View article: Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights
Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights Open
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 dua…
View article: A Helmholtz-type decomposition for the space of symmetric matrices
A Helmholtz-type decomposition for the space of symmetric matrices Open
In this paper, we introduce a Helmholtz-type decomposition for the space of square integrable, symmetric-matrix-valued functions analogous to the standard Helmholtz decomposition for vector fields. This decomposition provides a better unde…
View article: A T1 theorem for general Calder\'on-Zygmund operators with doubling weights, and optimal cancellation conditions, II
A T1 theorem for general Calder\'on-Zygmund operators with doubling weights, and optimal cancellation conditions, II Open
We extend the T1 theorem of David and Journ\'e, and the corresponding optimal cancellation conditions of Stein, to pairs of doubling measures.
View article: A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II
A weak to strong type T1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II Open
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 ty…
View article: Sum of squares I: scalar functions
Sum of squares I: scalar functions Open
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 …
View article: Sums of squares II: matrix functions
Sums of squares II: matrix functions Open
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 …
View article: A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications
A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications Open
View article: Continuity of infinitely degenerate weak solutions via the trace method
Continuity of infinitely degenerate weak solutions via the trace method Open
View article: Restricted Testing for Positive Operators
Restricted Testing for Positive Operators Open
View article: Control of the bilinear indicator cube testing property
Control of the bilinear indicator cube testing property Open
We show that the α-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure σ x ω is diagonally reverse doubling (in partic…
View article: Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients
Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients Open
View article: Weighted Alpert Wavelets
Weighted Alpert Wavelets Open
View article: A two weight local $Tb$ theorem for $n$-dimensional fractional singular integrals
A two weight local $Tb$ theorem for $n$-dimensional fractional singular integrals Open
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 …
View article: The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve
The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve Open
View article: A two weight local $Tb$ theorem for the Hilbert transform
A two weight local $Tb$ theorem for the Hilbert transform Open
We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T^{\alpha} on the real line \mathbb{R} , and any pair of locally finite positive Borel measures (\sigma,\omega) on \mathbb…
View article: A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications
A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with applications Open
Let $(X,d,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $μ$ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Bor…