Jonathan Bennett

University of Birmingham · School of Mathematics

We establish identities for the composition Tk,n(|gdσ^|2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{k,n}(|\widehat{gd\sigma }|^2)$$\end{document}, where g↦gdσ^\...

The Brascamp-Lieb inequalities are a generalization of the H\"older, Loomis-Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper we introduce an "adjoint" version of these inequalities, which can be viewed as an $L^p$ version of the entropy Brascamp-Lieb inequalities of Carlen a...

We establish identities for the composition $T_{k,n}(|\widehat{gd\sigma}|^2)$, where $g\mapsto \widehat{gd\sigma}$ is the Fourier extension operator associated with a general smooth $k$-dimensional submanifold of $\mathbb{R}^n$, and $T_{k,n}$ is the $k$-plane transform. Several connections to problems in Fourier restriction theory are presented.

In differential topology two smooth submanifolds $S_1$ and $S_2$ of euclidean space are said to be transverse if the tangent spaces at each common point together form a spanning set. The purpose of this article is to explore a much more general notion of transversality pertaining to a collection of submanifolds of euclidean space. In particular, we...

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-gener...

We explore the extent to which the Fourier transform of an Lp density supported on the sphere in Rn can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form X(|gdσ^|2) and R(|gdσ^|2), where X and R denote the X-ray and Radon transforms respectively; he...

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp-Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp-Lieb constants on (finitely-gener...

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature...

We explore the extent to which the Fourier transform of an $L^p$ density supported on the sphere in $\mathbb{R}^n$ can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $X(|\widehat{gd\sigma}|^2)$ and $\mathcal{R}(|\widehat{gd\sigma}|^2)$, where $X$...

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature...

We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic analysis and partial differential equations. Our proof consists of running an efficient, or "tight", induction on scales argument, which uses the existen...

We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant...

We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa an...

We prove that $$ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2, $$ where $u(x,t)$ is the solution to the linear time-dependent Schr\"odinger equation on $\mathbb{R}^2$ with initial datum $f$, and $X$ is the (spatial) X-ray transform on $\mathbb{R}^2$. In particular, we identify the best constant $C$ and show that a datum $f$ is an ex...

We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator $\rho$ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents $q$ and $r$, we find a characterisation of the exponents $\beta_+$ and $\beta_-$, exce...

Recent progress in multilinear harmonic analysis naturally raises questions about the local behaviour of the best constant (or bound) in the general Brascamp--Lieb inequality as a function of the underlying linear transformations. In this paper we prove that this constant is continuous, but is not in general differentiable.

Through the study of novel variants of the classical Littlewood-Paley-Stein $g$-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on $\mathbb{R}^d$ satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-...

The purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-soluti...

We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.

We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space.

The purpose of this article is to survey certain aspects of multilinear harmonic analysis related to notions of transversality. Particular emphasis will be placed on the multilinear restriction theory for the euclidean Fourier transform, multilinear oscillatory integrals, multilinear geometric inequalities, multilinear Radon-like transforms, and th...

We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schr$\"o$dinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we provide a generalization of their estimates in such a way that provides a unification with some sharp bilinear estimates prove...

We prove global versions of certain known nonlinear Brascamp–Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.

We prove local "L p -improving" estimates for a class of multilin-ear Radon-like transforms satisfying a strong transversality hypothesis. As a consequence, we obtain sharp multilinear convolution estimates for measures supported on fully transversal submanifolds of euclidean space of arbitrary dimension. We also prove global estimates for the same...

We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present a new approach to proving the non-endpoint cases using multilinear analysis.

We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of...

We consider the Schrödinger equation i∂ t u+Δu=0 with initial data in H s (ℝ n ). A classical problem is to identify the exponents s for which u(·,t) converges almost everywhere to the initial data as t tends to zero. In one spatial dimension, Carleson proved that the convergence is guaranteed when s=1 4, and Dahlberg and Kenig proved that divergen...

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\rightarrow L^q(\mathbb...

We refine results of Carleson, Sjögren and Sjölin regarding the pointwise convergence to the initial data of solutions to the Schrödinger equation. We bound the Hausdorff dimension of the sets on which convergence fails. For example, with initial data in H1(\mathbbR3){H^1(\mathbb{R}^{3})}, the sets of divergence have dimension at most one.

We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear Brascamp–Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru–Herr–Tataru and Bennett–Carbery–Wright.

We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣ s (ℝ n ). We control the weightedL 2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to that equation. Our partial positive results are complemented by some necess...

We discuss some heat-flow monotonicity phenomena pertaining to a variety of integral inequalities in Euclidean analysis.

We prove weighted local smoothing estimates for the resolvent of the Laplacian in three dimensions with weights belonging to the Kerman–Sawyer class. This class contains the well-known global Kato and Rollnik classes. We go on to discuss dispersive and Strichartz estimates for perturbations of the Laplacian by small potentials, and apply our result...

Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i s\Delta}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the free Schr\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow.

It is known that if q is an even integer, then the Lq(ℝd) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres ‘simultaneously slide’ to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These resu...

We study two problems closely related to each other. The first one is concerned with some smoothing weighted estimates with weights in a certain Morrey-Campanato spaces, for the solution of the free Schrödinger equation. The second one is a weighed trace inequality.

We prove that if u 1,u 2:(0,∞)×ℝd →(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝd then the function u:(0,∞)×ℝd →(0,∞) given by \(u^{1/p}=u_{1}^{1/p_{1}}*u_{2}^{1/p_{2}}\) also satisfies a heat inequality of a similar type provided \(\frac{1}{p_{1}}+\frac{1}{p_{2}}=1+\frac{1}{p}\) . On iterating, this result leads t...

We prove optimal radially weighted L2-norm inequalities for the Fourier extension operator associated to the unit sphere in n. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L 2 weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.

We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal Lp(𝕊2)→Lq(R𝕊2) estimates for the Fourier extension operator on large spher...

We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.

We give conditions on radial nonnegative weights \(W_1\) and \(W_2\) on \({\Bbb R}^n\), for which the a priori inequality $$\vert\vert(-\Delta_S)^{1/2}u\vert\vert_{L^2(W_1)}\leq C\vert\vert(\Delta+k^2)u\vert\vert_{L^2(W_2)}$$ holds with constant independent of \(k \in {\Bbb R}\). Here \(\Delta_S\) is the Laplace-Beltrami operator on the sphere \({\...

We prove d-linear analogues of the classical restriction and Kakeya conjectures in R d . Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called “joints...

We establish a diffeomorphism–invariant generalisation of the classical Loomis–Whitney inequality in R n . As a consequence we obtain a sharp trilinear restriction theorem for the Fourier transform in three dimensions.

A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties of Rogers and H\"older, Young, and Loomis-Whitney.

We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly add...

We establish a sharp trilinear inequality for the extension operator associated to the paraboloid in R 3 . Our proof relies on a recent generalisation of the classical Loomis–Whitney inequality.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on L2(double-struck S signn-1) × ... × L2(double-struck S sign n-1).

We provide sharp decay estimates for circular averages of a certain bilinear extension operator on L2(S1)×L2(S1).

We study the continuity properties of a projection derived from a recent characterization of Herglotz Wave Functions in the plane. Herglotz Wave Functions are the entire solutions of the Helmholtz equation which have L 2-Far-Field-Pattern. The behavior of this projection is reminiscent of the Disc Multiplier Operator on both L p and mixed L p -norm...

We explore decay estimates for L¹ circular means of the Fourier transform of a measure on R² in terms of its α -dimensional energy. We find new upper bounds for the decay exponent. We also prove sharp estimates for a certain family of randomised versions of this problem.

A generalised integral is used to obtain a Fourier multiplier relation for Calderón-Zygmund operators on L1(ℝn). In particular we conclude that an operator in our class is injective on L1 (ℝn) if it is injective on L2(ℝn).

A generalised integral is used to obtain a Fourier multiplier rela- tion for Calder on-Zygmund operators on L1(Rn). In particular we conclude that an operator in our class is injective on L1(Rn )i f it is injective on L 2 ( R n).