Some light on Littlewood-Paley theory
The purpose of this note is to correct an error in a paper of M. Cowling, G. Fendler and J.J.F. Fournier, and to give a counterexample to a conjecture of J.-L. Rubio de Francia.
math.CA↗
arXiv subjects
Terence Tao
Publications and source records attributed to Terence Tao.
Recent progress on the Kakeya conjecture
The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).
math.CA↗
Honeycombs and sums of Hermitian matrices
Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a {\em honeycomb}; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by ``classical vs. quantum'' analogies.
math.RT↗
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + εfor some absolute constant ε> 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call ``stickiness,'' ``planiness,'' and ``graininess.'' The purpose of this paper is to improve upon the known bounds for the Minkowski dimension of Besicovitch sets in three dimensions. As a by-product of the argument we obtain some strong conclusions on the structure of Besicovitch sets with almost-minimal Minkowski dimension.
math.CA↗
From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE
We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic progressions), oscillatory integrals (such as the Bochner-Riesz, restriction, and local smoothing problems), and the local and global well-posedness theory for non-linear dispersive and wave equations.
math.CA↗
A remark on circular means of Fourier transforms of measures
In a recent paper of Wolff the optimal decay of circular L^p means of compactly supported measures of finite energy was given for p>=2, with application to Falconer's distance problem. The question was then raised in that paper as to whether any non-trivial improvement to these estimates were available for p < 2. We answer this question in the negative.
math.CA↗
Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates
Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp $L^p$ null form estimates.
math.CA↗
An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension
We use geometrical combinatorics arguments, including the ``hairbrush'' and x-ray arguments of Wolff and the sticky/plany/grainy analysis of Katz, Laba, and Tao, to show that Besicovitch sets in R^n have Minkowski dimension at least (n+2)/2 + \eps_n for all n > 3, where \eps_n > 0 is an absolute constant depending only on n. This complements the results of Katz, Laba, and Tao, which established the same result for n=3, and of Bourgain and Katz-Tao, arithmetic combinatorics techniques to establish the result for n > 8. In contrast to previous work, our arguments will be purely geometric and do not require arithmetic combinatorics.
math.CA↗
A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture
Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that $# \{a+2b: (a,b) \in G\} \leq N$. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least $4n/7 + 3/7$. This is new for $n > 8$.
math.CO↗
Endpoint multiplier theorems of Marcinkiewicz type
We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty} and L \log^{1/2} L to L^{1,\infty}, and that these estimates are sharp.
math.CA↗
Sharp Lorentz space estimates for rough operators
We demonstrate the $(H^1,L^{1,2})$ or $(L^p,L^{p,2})$ mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number.
math.CA↗
A converse extrapolation theorem for translation invariant operators
We prove the converse of Yano's extrapolation theorem for translation invariant operators.
math.FA↗
Convex bodies with a point of curvature do not have Fourier bases
We prove that no smooth symmetric convex body $Ω$ with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case was proven by Kolountzakis). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if $Ω$ can tile $R^d$ by translations.
math.CA↗
Multi-linear operators given by singular multipliers
We prove L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer, as well as the bilinear Hilbert transform.
math.CA↗
The weak-type (1,1) of L \log L homogeneous convolution operators
We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the result of Seeger. The proof is in a similar spirit to that of Christ and Rubio de Francia.
math.CA↗
An x-ray estimate in $R^n$
We prove an x-ray estimate in general dimension which is stronger than the Kakeya estimates of Wolff. This generalizes an x-ray estimate in three dimensions which is also due to Wolff.
math.CA↗
Ill-posedness for one-dimensional wave maps at the critical regularity
We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(f Dg)$ in these spaces.
math.AP↗
The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture
We introduce the honeycomb model of BZ polytopes, which calculate Littlewood-Richardson coefficients, the tensor product rule for GL(n). Our main result is the existence of a particularly well-behaved honeycomb with given boundary conditions (choice of triple of representations to be tensored together). This honeycomb is necessarily integral, which proves the "saturation conjecture", extending results of Klyachko to give a complete answer to which L-R coefficients are positive. This in turn has as a consequence Horn's conjecture from 1962 characterizing the spectrum of the sum of two Hermitian matrices.
math.RT↗