arXiv subjects
Publications and source records attributed to Terence Tao.
We show that if $f$ is locally in $L\log\log L$ then the lacunary spherical means converge almost everywhere. The argument given here is a model case for more general results on singular maximal functions and Radon transforms (see ref. 6).
We show that some singular maximal functions and singular Radon transforms satisfy a weak type $L\log\log L$ inequality. Examples include the maximal function and Hilbert transform associated to averages along a parabola. The weak type inequality yields pointwise convergence results for functions which are locally in $L\log\log L$.
We consider a basic $d$-adic model for the scattering transform on the line. We prove $L^2$ bounds for this scattering transform and a weak $L^2$ bound for a Carleson type maximal operator. The latter implies boundedness of $d$-adic models of generalized eigenfunctions of Dirac type operators with potential in $L^2(R)$. We show that this result cannot be obtained by estimating the terms in the natural multilinear expansion of the scattering transform.
Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $λ$. Let $ψ$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of $ψ$ will grow as $λ^{1/2}$ as $λ\to \infty$. We prove an upper bound of the form $\|ψ\|_2^2 \leq Cλ$ for any Riemannian manifold, and a lower bound $c λ\leq \|ψ\|_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $λ$. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate.
Christ and Kiselev have established that the generalized eigenfunctions of one-dimensional Dirac operators with $L^p$ potential $F$ are bounded for almost all energies for $p < 2$. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series $\sum_n T_n(F, ..., F)$ and carefully bounding each term $T_n(F, ..., F)$. It is conjectured that the results of Christ and Kiselev also hold for $L^2$ potentials $F$. However in this note we show that the bilinear term $T_2(F,F)$ and the trilinear term $T_3(F,F,F)$ are badly behaved on $L^2$, which seems to indicate that multilinear expansions are not the right tool for tackling this endpoint case.
We prove $L^p$ estimates for the Walsh model of the maximal bi-Carleson operator (which is a hybrid of the bilinear Hilbert transform and the Carleson maximal operator which appears naturally in the eigenfunction problem for one-dimensional Schrodinger operators). The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.
The set of possible spectra (λ,μ,ν) of zero-sum triples of Hermitian matrices forms a polyhedral cone. We give a complete determination of its facets, finishing a long story with recent highlights by [Helmke-Rosenthal, Klyachko, Belkale]. We introduce_puzzles_, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and will probably be the main point of interest for many readers. As the proofs indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results in [H,Kly,HR,Be]. Along the way we give a characterization of ``rigid'' puzzles, which we use to prove a conjecture of W. Fulton: ``if for a triple of dominant weights λ,μ,νof GL(n,C) the irreducible representation V_νappears exactly once in V_λtensor V_μ, then for all N\in \naturals, V_{Nλ} appears exactly once in V_{Nλ} tensor V_{Nμ}.''
Multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak $L^q$ estimate for a single index $q$ (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak $L^q$ estimate. Under this assumption, in this expository note we give a general multilinear interpolation theorem which allows one to obtain strong type boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case $q \leq 1$. When $q > 1$, weak $L^q$ has a predual, and such strong type boundedness can be easily obtained by duality and multilinear interpolation.
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or with connections with non-zero curvature). In this note we give some techniques to prove these estimates which rely more on physical space methods such as vector fields, tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of recent papers of Wolff).
We prove uniform $L^p$ bounds for multilinear operators which are given by multipliers whose symbols are singular on a one dimensional subspace. The novelty is that these bounds are uniform in the choice of the subspace.
We prove uniform $L^p$ estimates for a family of paraproducts and corresponding maximal operators.
We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.
We prove L^p estimates for the Walsh model of the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.
We prove L^p estimates for the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. In a previous paper these estimates were obtained for a simpler Walsh model for this operator, but in the Fourier case additional complications arise due to the inability to perfectly localize in both space and frequency.
In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $δ$-discretized versions of these conjectures and show that in a certain sense that these discretized versions are equivalent. In particular, it appears that to progress on any of these problems one must prove a quantitative statement about the existence of sub-rings of $R$ of dimension 1/2.
We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.
We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.