arXiv subjects
Publications and source records attributed to Anthony Carbery.
We prove that a local $L^p(V_2)$ variation norm estimate fails for circular means in two dimensions, and quantify this failure by proving lower bounds for functions of exponential type. This is related to lower bounds for Fourier multipliers supported on annuli, of the type considered by C\'ordoba.
Littlewood--Paley theory began with the classic paper of Littlewood and Paley (J.\ E.\ Littlewood, R.\ E.\ A.\ C.\ Paley, {\em Theorems on Fourier Series and Power Series}. J. Lond. Math. Soc. (1), {\bf 6} (1931), 230--33). We discuss this paper and its impact from a historical perspective. We include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.
Let $\Gamma$ be a compact patch of a well-curved $C^{n+1}$ curve in $\mathbb{R}^n$ with induced Lebesgue measure ${\rm d} \lambda$, and let $g \mapsto \widehat{g \,{\rm d}\lambda}$ be the Fourier extension operator for $\Gamma$. Then we have, for arbitrary non-negative weights $w$, \begin{equation*} \int_{B_R} |\widehat{g \,{\rm d}\lambda}|^2w \leq C_{n,a} R^{a} \sup_S \left(\int_S w\right)\int_\Gamma |g|^2 \, {\rm d} \lambda \end{equation*} for any $a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$, where the $\sup$ is over all $1$-neighbourhoods $S$ of hyperplanes whose normals are parallel to the tangent at some point of $\Gamma$. This represents partial progress on the Mizohata-Takeuchi conjecture for curves in dimensions $n \geq 3$, improving upon the exponent $a=n-1$ which can be obtained as a consequence of the Agmon-H\"ormander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.
We consider some integral-geometric quantities that have recently arisen in harmonic analysis and elsewhere, derive some sharp geometric inequalities relating them, and place them in a wider context.
Let $\Sigma$ be a strictly convex, compact patch of a $C^2$ hypersurface in $\mathbb{R}^n$, with non-vanishing Gaussian curvature and surface measure $d\sigma$ induced by the Lebesgue measure in $\mathbb{R}^n$. The Mizohata--Takeuchi conjecture states that \begin{equation*} \int |\widehat{gd\sigma}|^2w \leq C \|Xw\|_\infty \int |g|^2 \end{equation*} for all $g\in L^2(\Sigma)$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$, where $X$ denotes the $X$-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every $\epsilon>0$, there exists a positive constant $C_\epsilon$, which depends only on $\Sigma$ and $\epsilon$, such that for all $R \geq 1$ and all weights $w:\mathbb{R}^n\rightarrow [0,+\infty)$ we have \begin{equation*} \int_{B_R} |\widehat{gd\sigma}|^2w \leq C_\epsilon R^\epsilon \sup_T \left(\int _T w^{\frac{n+1}{2}}\right)^{\frac{2}{n+1}}\int |g|^2, \end{equation*} where $T$ ranges over the family of all tubes in $\mathbb{R}^n$ of dimensions $R^{1/2} \times \dots \times R^{1/2} \times R$. From this we deduce the Mizohata--Takeuchi conjecture with an $R^{\frac{n-1}{n+1}}$-loss; i.e., that \begin{equation*} \int_{B_R} |\widehat{gd\sigma}|^2w \leq C_\epsilon R^{\frac{n-1}{n+1}+ \epsilon}\|Xw\|_\infty\int |g|^2 \end{equation*} for any ball $B_R$ of radius $R$ and any $\epsilon>0$. The power $(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of $\widehat{gd\sigma}$ beyond 'decoupling axioms' are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.
We develop a framework for a duality theory for general multilinear operators which extends that for transversal multilinear operators which has been established in arXiv:1809.02449. We apply it to the setting of joints and multijoints, and obtain a "factorisation" theorem which provides an analogue in the discrete setting of results of Bourgain and Guth (arXiv:0811.2251 and arXiv:1012.3760) from the Euclidean setting.
The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovski\u{\i}'s factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers. In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved - it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida-Hewitt theory of such measures - the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.
In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators. Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitative estimates on each operator separately. The concept of disentanglement recurs throughout the paper. The methods we used in the previous work - principally convex optimisation - relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.
We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors $w_i$ of a not necessarily orthonormal basis of $\mathbb{R}^n$, or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors $w_i$. Restricted and functional versions of the inequality will also be considered.
Let $\mathcal{L}$ be a family of lines and let $\mathcal{P}$ be a family of $k$-planes in $\mathbb{F}^n$ where $\mathbb{F}$ is a field. In our first result we show that the number of joints formed by a $k$-plane in $\mathcal{P}$ together with $(n-k)$ lines in $\mathcal{L}$ is $O_n(|\mathcal{L}||\mathcal{P}|^{1/(n-k)}$). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $\mathbb{F}$. In contrast, for our second result, we work in the three-dimensional Euclidean space $\mathbb{R}^3$, and we establish the Kakeya-type estimate \begin{equation*}\sum_{x \in J} \left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim |\mathcal{L}|^{3/2}\end{equation*} where $J$ is the set of joints formed by $\mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality.
We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey-Nikisin-Stein theory of factorisation of operators, and present a fully multilinear version of Maurey's fundamental theorem on factorisation of operators through $L^1$. The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of $L^\infty$, and the Yosida-Hewitt theory of finitely additive measures. We consider the connections of the theory with the theory of interpolation of operators. We discuss the ramifications of the theory in the context of concrete families of geometric inequalities, including Loomis-Whitney inequalities, Brascamp-Lieb inequalities and multilinear Kakeya inequalities.
The notion of the magnitude of a metric space was introduced by Leinster in [8] and developed in [10], [9], [11] and [16], but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton [9] in the special case of balls in dimension three. In general we show that the magnitude of an odd-dimensional ball is a rational function of its radius. In addition to Fourier-analytic and PDE techniques, the arguments also involve some combinatorial considerations.
We show that a certain conjectured optimal reverse Littlewood- Paley inequality would, if true, imply sharp results for the Kakeya maximal function, the Bochner-Riesz means and the Fourier restriction operator.
We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodrán result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.
Let L_1, ..., L_d be pairwise disjoint collections of lines in a d-dimensional vector space over some field. If the collections are sufficiently generic we prove that there exists a d-colouring of the set of multijoints J such that for each j, for each line in L_j, the number of points on it of colour j is O(|J|^{1/d}).
We initiate the study of the finiteness condition $\int_Ωu(x)^{-β}\,dx\leq C(Ω,β)<+\infty$ where $Ω\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $Δu=-1$ in $Ω$, $u=0$ on $\partialΩ$. The central issue which we address is that of determining the range of values of the parameter $β>0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $Ω$ and demands on the nature of the constant $C(Ω,β)$. Classes of domains for which our analysis applies include bounded piecewise $C^1$ domains in ${\mathbb{R}}^n$, $n\geq 2$, with conical singularities (in particular polygonal domains in the plane), polyhedra in ${\mathbb{R}}^3$, and bounded domains which are locally of class $C^2$ and which have (finitely many) outwardly pointing cusps. For example, we show that if $u_N$ is the solution of the Saint Venant problem in the regular polygon $Ω_N$ with $N$ sides circumscribed by the unit disc in the plane, then for each $β\in(0,1)$ the following asymptotic formula holds: % {eqnarray*} \int_{Ω_N}u_N(x)^{-β}\,dx=\frac{4^βπ}{1-β} +{\mathcal{O}}(N^{β-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions $v$ satisfying $v(0)=0$, $\nabla v(0)=0$ and $Δv\geq c>0$.
We give an essentially self-contained proof of Guth's recent endpoint multilinear Kakeya theorem which avoids the use of somewhat sophisticated algebraic topology, and which instead appeals to the Borsuk-Ulam theorem.
These informal notes, not intended for publication, provide an approach to the Borsuk--Ulam theorem via Stokes' theorem, in a similar spirit to Lima's proof of the Brouwer fixed point theorem. They are intended to be accessible to anyone with a knowledge of basic point-set topology and several variable calculus. Some of the argument relies upon previous ideas of Shchepin.