arXiv subjects

Philip T. Gressman

Publications and source records attributed to Philip T. Gressman.

At least 19 recordsLinked to original sources

arXiv2025

Finiteness of the Hölder-Brascamp-Lieb Constant Revisited

Abstract Hölder-Brascamp-Lieb inequalities have become a ubiquitous tool in Fourier analysis in recent years, due in large part to a theorem of Bennett, Carbery, Christ, and Tao (2008,2010) characterizing finiteness of the Hölder-Brascamp-Lieb constant. Here we provide a new characterization of a substantially different nature involving directed graphs of subspaces. Its practical value derives from its complementary nature to the Bennett, Carbery, Christ, and Tao conditions: it creates a means by which one can establish finiteness of the Hölder-Brascamp-Lieb constant by analysis of a well-chosen, finite list of subspaces rather than by checking conditions on all subspaces of the underlying vector space. The proof is elementary and is essentially an "explicitization" of the semi-explicit factorization algorithm of Carbery, Hänninen, and Valdimarsson (2023).

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arXiv2024

Generalized Sublevel Estimates for Form-Valued Functions and Related Results for Radon-like Transforms

Motivated by the testing condition for Radon-Brascamp-Lieb multilinear functionals established in arXiv:2201.12201, this paper is concerned with identifying local conditions on smooth maps $u(t)$ with values in the space of decomposable p-forms on some real vector space V which guarantee uniform integrability of $||u(t)||^{-τ}$ over a certain natural, noncompact family of norms. One can loosely regard this problem as a higher-dimensional analogue of establishing uniform bounds for the size of a sublevel set of a function in terms of the size of its derivatives. The resulting theorem relies extensively on ideas from Geometric Invariant Theory to understand what appropriate derivative bounds look like in this context. Several examples and applications are presented, including a new local characterization of so-called "model" Radon-like transforms in terms of the semistability of a natural curvature functional (giving an equivalent but rather different criterion than the one first established in arXiv:2303.03325).

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arXiv2023

Local curvature of maximally nondegenerate Radon-like transforms

This paper gives a complete geometric characterization in all dimensions and codimensions of those Radon-like transforms which, up to endpoints, satisfy the largest possible range of local $L^p \rightarrow L^q$ inequalities permitted by quadratic-type scaling. The necessary and sufficient curvature-type criterion is phrased in terms of an associated Newton-like diagram. In the case of averages over families of polynomial graphs, the curvature condition implies sharp endpoint estimates as well. The proof relies on the recently-developed multilinear Radon-Brascamp-Lieb testing criterion and a refined version of differential inequalities for polynomials first appearing in work on the Oberlin affine curvature condition.

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arXiv2022

A new type of superorthogonality

We provide a simple criterion on a family of functions that implies a square function estimate on $L^p$ for every even integer $p \geq 2$. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known.

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arXiv2022

On the strict majorant property in arbitrary dimensions

In this work we study $d$-dimensional majorant properties. We prove that a set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least $d+2$ frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $p \not\in 2 {\mathbb N}$ of length 2. Any infinite set of frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $p \not\in 2 {\mathbb N}$ of length $2$. Finally, given any $p>0$ with $p \not\in 2{\mathbb N}$, we exhibit a set of $d+2$ frequencies on the moment curve in ${\mathbb R}^d$ that violate the strict majorant property on $L^p([0,1]^d).$

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arXiv2022

On frames of smooth, compactly-supported wave packets adapted to tilings of frequency space

We establish a broad notion of admissible tilings of frequency space which admit associated wave packet frames with elements which are smooth and compactly supported. The framework is designed to allow for tile geometries which are minimally constrained by the need to accommodate Schwartz tails on the Fourier side and goes beyond the usual scale of geometries ranging from Gabor to wavelet-type decompositions. The approach builds on techniques of Hernández, Labate and Weiss and Labate, Weiss, and Wilson as well as a classical result of Ingham characterizing the best-possible Fourier decay for functions of compact support.

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arXiv2020

$L^p$-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality

This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension $n$ of the ambient space and the dimension $k$ of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang and extended by Zorin-Kranich, and on recent results for geometric nonconcentration inequalities. As an initial application of this new approach, this paper establishes sharp restricted strong type $L^p$-improving inequalities for certain model quadratic submanifolds in the range $k < n \leq 2k$.

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arXiv2020

Simulating COVID-19 in a University Environment

Residential colleges and universities face unique challenges in providing in-person instruction during the COVID-19 pandemic. Administrators are currently faced with decisions about whether to open during the pandemic and what modifications of their normal operations might be necessary to protect students, faculty and staff. There is little information, however, on what measures are likely to be most effective and whether existing interventions could contain the spread of an outbreak on campus. We develop a full-scale stochastic agent-based model to determine whether in-person instruction could safely continue during the pandemic and evaluate the necessity of various interventions. Simulation results indicate that large scale randomized testing, contact-tracing, and quarantining are important components of a successful strategy for containing campus outbreaks. High test specificity is critical for keeping the size of the quarantine population manageable. Moving the largest classes online is also crucial for controlling both the size of outbreaks and the number of students in quarantine. Increased residential exposure can significantly impact the size of an outbreak, but it is likely more important to control non-residential social exposure among students. Finally, necessarily high quarantine rates even in controlled outbreaks imply significant absenteeism, indicating a need to plan for remote instruction of quarantined students.

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arXiv2019

Multilinear oscillatory integral operators and geometric stability

In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood-Paley decompositions. This decomposition very nearly diagonalizes the problem and seems likely to have useful applications to other geometrically-inspired objects in Fourier analysis.

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arXiv2019

Reversing a philosophy: from counting to square functions and decoupling

Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality. Second, in our main result we prove an $L^{2n}$ square function estimate (which implies a corresponding decoupling estimate) for the extension operator associated to a non-degenerate curve in $\mathbb{R}^n$. The proof is via a combinatorial argument that builds on the idea that if $γ$ is a non-degenerate curve in $\mathbb{R}^n$, then as long as $x_1,\ldots, x_{2n}$ are chosen from a sufficiently well-separated set, then $ γ(x_1)+\cdots+γ(x_n) = γ(x_{n+1}) + \cdots + γ(x_{2n}) $ essentially only admits solutions in which $x_1,\ldots,x_n$ is a permutation of $x_{n+1},\ldots, x_{2n}$.

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arXiv2017

On the Oberlin affine curvature condition

In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb R^n$ , $1 \leq d \leq n-1$. We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible. The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result of the author concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

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arXiv2016

Higher decay inequalities for multilinear oscillatory integrals

In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each $p_j \geq 2$, expanding the known results for this operator well outside the previous range $\sum_{j=1}^d p_j^{-1} = d-1$. Our theorem assumes second-order nondegeneracy condition of Varchenko type, and as a corollary reproduces Varchenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb R^d$.

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arXiv2016

Generalized curvature for certain Radon-like operators of intermediate dimension

This paper establishes $L^p$-improving estimates for a variety of Radon-like transforms which integrate functions over submanifolds of intermediate dimension. In each case, the results rely on a unique notion of curvature which relates to, but is distinct from, Phong-Stein rotational curvature. The results obtained are sharp up to the loss of endpoints. The methods used are a new adaptation of the familiar method of inflation developed by Christ and others. Unlike most previous instances of this method, the present application does not require any particular linear algebraic relations to hold for the dimension and codimension.

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arXiv2016

Fourier extension for extremal quadratic submanifolds

This note establishes the full range of $L^p$--$L^q$ Fourier extension estimates for the model $n$-dimensional quadratic submanifold in ${\mathbb R}^{n(n+3)/2}$ parametrized by $γ(x_1,\ldots,x_n) := (x_1,\ldots,x_n, (x_i x_j)_{1 \leq i \leq j \leq n})$. This class of submanifolds is extremal in the sense that an $n$-dimensional quadratic submanifold of ${\mathbb R}^d$ can only satisfy nontrivial Fourier extension inequalities when $d \leq \frac{n(n+3)}{2}$. The proof is via an inflation-type argument, with the unexpected twist that a significant amount of "overinflation" is necessary but in no way limits the sharpness of the argument.

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arXiv2015

Maximal Decay Inequalities for Trilinear oscillatory integrals of convolution type

In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.

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arXiv2014

Damping oscillatory integrals by the Hessian determinant via Schrödinger

We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the Hessian determinant of the phase is bounded below. We show in two dimensions that the desired outcome is not always possible, but does occur for a broad class of phases which may be described in terms of the Newton polygon. The estimates obtained are uniform with respect to linear perturbation of the phase and uniform in the cutoff value of the Hessian determinant. In the course of the proof, we investigate a geometrically-invariant approach to making uniform estimates of qualitatively nondegenerate oscillatory integrals. The approach illuminates a previously unknown, fundamental relationship between the asymptotics of oscillatory integrals and the Schrödinger equation.

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arXiv2013

An operator van der Corput estimate arising from oscillatory Riemann-Hilbert problems

We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann-Hilbert problems associated with partial differential equations in the Ablowitz-Kaup-Newell-Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author, the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can simultaneously accommodate phases with low regularity (as low as $C^{1,α}$), local singularities, and essentially arbitrary sets of stationary points that degenerate to finite or infinite order.

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arXiv2013

$L^p$-nondegenerate Radon-like operators with vanishing rotational curvature

We consider the $L^p \rightarrow L^q$ mapping properties of a model family of Radon-like operators integrating functions over n-dimensional submanifolds of ${\mathbb R}^{2n}$. It is shown that nonvanishing rotational curvature is never generic when $n \geq 2$ and is, in fact, impossible for all but finitely many values of $n$. Nevertheless, operators satisfying the same $L^p \rightarrow L^q$ estimates as the "nondegenerate" case (modulo the endpoint) are dense in the model family for all $n$.

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