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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 $||u(t)||^{-\tau}$ 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)., Comment: 66 pages

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., Comment: 58 pages

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., Comment: 12 pages

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 Hern\'{a}ndez, 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., Comment: 20 pages

This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in one direction and on a careful analysis of Knapp-type examples in the other. Various applications are discussed which demonstrate new and subtle interplay between curvature and transversality and establish nontrivial mixed-norm $L^p$-improving inequalities in the model case of convolution with affine hypersurface measure on the paraboloid., Comment: 60 pages

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 $L^p([0,1]^d).$, Comment: 22 pages

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 $k < n \leq 2k$., Comment: 53 pages, 2 figures

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., Comment: 30 pages, 9 figures; fixed minor typos and rephrased some unclear points

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., Comment: 21 pages

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 $\gamma$ 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 $ \gamma(x_1)+\cdots+\gamma(x_n) = \gamma(x_{n+1}) + \cdots + \gamma(x_{2n}) $ essentially only admits solutions in which $x_1,\ldots,x_n$ is a permutation of $x_{n+1},\ldots, x_{2n}$., Comment: 16 pages

This paper is devoted to a systematic study of certain geometric integral inequalities which arise in continuum combinatorial approaches to $L^p$-improving inequalities for Radon-like transforms over polynomial submanifolds of intermediate dimension. The desired inequalities relate to and extend a number of important results in geometric measure theory., Comment: 42 pages

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., Comment: 28 pages, 2 figures

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 $\mathbb R^d$., Comment: 12 pages

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., Comment: 43 pages

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 $\gamma(x_1,\ldots,x_n) := (x_1,\ldots,x_n, (x_i x_j)_{1 \leq i \leq j \leq n})$. 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., Comment: This paper has been withdrawn by the author as this problem has been previously solved by D. M. Oberlin (Canad. Math. Bull. Vol. 48 (2), 2005 pp. 260--266) using very similar methods

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.

We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calder\'on commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form., Comment: 12 pages

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 Schr\"odinger equation., Comment: 19 pages

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 $n$., Comment: 11 pages

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 $C^{1,\alpha}$), local singularities, and essentially arbitrary sets of stationary points that degenerate to finite or infinite order., Comment: 27 pages

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9]., Comment: 20 pages

We consider a generalization of the notion of spaces of homogeneous type, inspired by recent work of Street [21] on the multi-parameter Carnot-Caratheodory geometry, which imbues such spaces with differentiability structure. The setting allows one to formulate estimates for scalar oscillatory integrals on these spaces which are uniform and respect the underlying geometry of both the space and the phase function. As a corollary we obtain a generalization of a theorem of Bruna, Nagel, and Wainger [1] on the asymptotic behavior of scalar oscillatory integrals with smooth, convex phase of finite type., Comment: 31 pages

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. $$ We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \cite{ksLM} to include all $\alpha\in (0, 74/75)$., Comment: 6 pages, to appear in Advances in Mathematics

This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial perturbations of the Maxwellian equilibrium states, as announced in \cite{gsNonCutA}. We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -n$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $\gamma \ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma \ge -2s$, as conjectured in Mouhot-Strain \cite{MR2322149}. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory., Comment: This paper is a combination, simplification, and extension of two separate preprints originally posted on the arXiv as (arXiv:0912.0888v1) and (arXiv:1002.3639v1). It was revised in July 2010 for the referee. In particular we now estimate all $\gamma > -n$. Please cite this version. 77 pages

This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin, relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new $L^p$-improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger for FIOs (building on the ideas of Sogge and Mockenhaupt, Seeger, and Sogge); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context., Comment: 40 pages

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$) in the trilinear $L^2(\R^n)$ energy $<\mathcal{Q}(g,f),f>$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(\R^n)$., Comment: 29 pages, updated file based on referee report; Advances in Mathematics (2011)

In this work we prove global stability for the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $\gamma + 2s \ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma + 2s < 0$, our solutions decay polynomially fast in time with any rate. These results are constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma + 2s \ge 0$, as conjectured in Mouhot-Strain (2007)., Comment: This file has not changed, but this work has been combined with (arXiv:0912.0888v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v1

We prove the existence and exponential decay of global in time strong solutions to the Boltzmann equation without any angular cut-off, i.e., for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p>3$, and more generally, the full range of angular singularities $s=\nu/2 \in(0,1)$. These appear to be the first unique global solutions to this fundamentally important model, which grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects of grazing collisions in the Boltzmann theory., Comment: This file has not changed, but this work has been combined with (arXiv:1002.3639v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v1

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008)., Comment: 14 pages

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators which generalize mixed derivatives in $\R^d$. As a consequence, it is shown that, in the case of both of these previous works, improved uniform decay rates are possible in many situations., Comment: 16 pages, 1 figure

In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible. Inequalities of this type are then applied to extend the results of Tao and Wright (JAMS, 2003) to obtain endpoint restricted weak-type estimates for averages over curves given by polynomials., Comment: 21 pages

This paper establishes endpoint $L^p-L^q$ and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik, and Tang concerning oscillatory integral operators., Comment: 32 pages, 2 figures

This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs., Comment: 22 pages; v2 added references

In this paper, optimal $L^p-L^q$ estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the $k$-plane transform. An important advance over previous work is that full $L^p-L^q$ estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is lead to make coercivity estimates for certain functionals on $L^p$ for $p < 1$., Comment: 23 pages; 1 figure

This paper reports on our effort in modeling realistic astrophysical neutron star binaries in general relativity. We analyze under what conditions the conformally flat quasiequilibrium (CFQE) approach can generate ``astrophysically relevant'' initial data, by developing an analysis that determines the violation of the CFQE approximation in the evolution of the binary described by the full Einstein theory. We show that the CFQE assumptions significantly violate the Einstein field equations for corotating neutron stars at orbital separations nearly double that of the innermost stable circular orbit (ISCO) separation, thus calling into question the astrophysical relevance of the ISCO determined in the CFQE approach. With the need to start numerical simulations at large orbital separation in mind, we push for stable and long term integrations of the full Einstein equations for the binary neutron star system. We demonstrate the stability of our numerical treatment and analyze the stringent requirements on resolution and size of the computational domain for an accurate simulation of the system., Comment: 22 pages, 18 figures, accepted to Phys. Rev. D

We study the dynamical evolution of a large amplitude r-mode by numerical simulations. R-modes in neutron stars are unstable growing modes, driven by gravitational radiation reaction. In these simulations, r-modes of amplitude unity or above are destroyed by a catastrophic decay: A large amplitude r-mode gradually leaks energy into other fluid modes, which in turn act nonlinearly with the r-mode, leading to the onset of the rapid decay. As a result the r-mode suddenly breaks down into a differentially rotating configuration. The catastrophic decay does not appear to be related to shock waves at the star's surface. The limit it imposes on the r-mode amplitude is significantly smaller than that suggested by previous fully nonlinear numerical simulations., Comment: Published in Phys. Rev. D Rapid Comm. 66, 041303(R) (2002)

It has been conjectured that in head-on collisions of neutron stars (NSs), the merged object would not collapse promptly even if the total mass is higher than the maximum stable mass of a cold NS. In this paper, we show that the reverse is true: even if the total mass is {\it less} than the maximum stable mass, the merged object can collapse promptly. We demonstrate this for the case of NSs with a realistic equation of state (the Lattimer-Swesty EOS) in head-on {\it and} near head-on collisions. We propose a ``Prompt Collapse Conjecture'' for a generic NS EOS for head on and near head-on collisions., Comment: 5 pages, 7 figures, submitted to Phy. Rev. D Rapid Comm