On multilinear oscillatory integrals, nonsingular and singular (2009)
Michael Christ, Xiaochun Li, Terence Tao, Christoph Thiele
Abstract. Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. L p norm inequalities are established for multilinear integral...
Jonathan Bennett, Anthony Carbery, Michael Christ, Terence Tao
Abstract. 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 Hölder, Young, and...
A finitary version of Gromov's polynomial growth theorem (2009)
We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $ R_0 > \exp(\exp(Cd^C))$ for which the number of...
Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm (2009)
Keel, Markus, Roy, Tristan, Tao, Terence
We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of...
A variation norm Carleson theorem (2009)
Oberlin, Richard, Seeger, Andreas, Tao, Terence, Thiele, Christoph, Wright, James
We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are...
Abstract. We prove new velocity averaging results for second-order multidimensional equations of the general form, L(∇x,v)f(x, v) =g(x, v) where L(∇x,v):=a(v) ·∇x −∇ ⊤ x · b(v)∇x....
A remark on partial sums involving the Mobius function (2009)
Let $ \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the...
Random matrices: Universality of local eigenvalue statistics up to the edge (2009)
This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the...
Global regularity of wave maps VII. Control of delocalised or dispersed solutions (2009)
This is the final paper in the series \cite{tao:heatwave}, \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that establishes global regularity for two-dimensional wave maps into...
Operator splitting for the KdV equation (2009)
Holden, Helge, Karlsen, Kenneth H., Risebro, Nils Henrik, Tao, Terence
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$ where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg-de Vries...
Bulk universality for Wigner hermitian matrices with subexponential decay (2009)
Erdos, Laszlo, Ramirez, Jose, Schlein, Benjamin, Tao, Terence, Vu, Van, Yau, Horng-Tzer
We consider the ensemble of $n \times n$ Wigner hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar...
Sumset and inverse sumset theorems for Shannon entropy (2009)
Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated...
Freiman's theorem for solvable groups (2009)
Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised...
Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation (2009)
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u:...
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm (2009)
We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed...
Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions (2009)
In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic space $\H^m$ was reduced to the problem of...
Random matrices: Universality of local eigenvalue statistics (2009)
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the...
An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation (2009)
A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two...
The Power of Convex Relaxation: Near-Optimal Matrix Completion (2009)
Candes, Emmanuel J., Tao, Terence
This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of...
Random matrices: The distribution of the smallest singular values (2009)
Let $\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the...
Restriction and Kakeya phenomena for finite fields (2009)
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations...
The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation (2009)
We investigate the behaviour of solutions $\phi = \phi^{(p)}$ to the one-dimensional nonlinear wave equation $-\phi_{tt} + \phi_{xx} = -|\phi|^{p-1} \phi$ with initial data $\phi(0,x) = \phi_0(x)$,...
An inverse theorem for the uniformity seminorms associated with the action of $F^\omega$ (2009)
Bergelson, Vitaly, Tao, Terence, Ziegler, Tamar
Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^\omega}$ of the infinite abelian...
New bounds for Szemeredi's theorem, I: progressions of length 4 in finite field geometries (2009)
Let k ≥ 3 be an integer, and let G be a finite abelian group with |G| = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A ⊆ G which does not contain k...
A quantitative version of the Besicovitch projection theorem via multiscale analysis (2009)
By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has...
To Laura, for being so patient. Contents Preface ix (2008)
and global analysis of nonlinear dispersive
Inverse Littlewood-Offord theorems and the condition number of random discrete matrices (2008)
Abstract. Consider a random sum η1v1 +... + ηnvn, where η1,..., ηn are i.i.d. random signs and v1,..., vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities...
Puzzles and (equivariant) cohomology of Grassmannians (2008)
The product of two Schubert cohomology classes on a Grassmannian Grk(C n) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now...
To Laura, for being so patient. Contents Preface ix (2008)
1.1. General theory 2 1.2. Gronwall’s inequality 11
Emmanuel J. C, Justin Romberg, Terence Tao
Abstract—This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal and a randomly chosen set of frequencies. Is it...
The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm...
The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the...
L 2 BOUNDS FOR NORMAL DERIVATIVES OF DIRICHLET EIGENFUNCTIONS (2008)
Abstract. 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...
Using the harmonic map heat flow and the function spaces of Tataru and the author, we establish a large data local well-posedness result in the energy class for wave maps from two-dimensional...
Random matrices: Universality of ESDs and the circular law (2008)
Tao, Terence, Vu, Van, Krishnapur, Manjunath
Given an $n \times n$ complex matrix $A$, let $$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x, \Im \lambda_i \le y\}|$$ be the empirical spectral distribution (ESD) of its...
We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$ in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx <...
Classification of Almost Quarter-Pinched Manifolds (2008)
We show that if a simply connected manifold is almost quarter pinched then it is diffeomorphic to a CROSS (a compact rank one symmetric space) or a sphere.
The Mobius function is strongly orthogonal to nilsequences (2008)
We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and...
Using the harmonic map heat flow, we construct an energy class for wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a...
The sum-product phenomenon in arbitrary rings (2008)
The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite...
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008)
We show that wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable...
Random matrices: A general approach for the least singular value problem (2008)
Let $x$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $x$ and $M$ be an arbitrary matrix. We give a...
We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schr\"odinger equations $i u_t + \Delta u = |u|^{p-1} u...
A quantitative ergodic theory proof of Szemerédi’s theorem (2008)
A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions...
Applied and Computational (2008)
Emmanuel Candes, Mark Rudelson, Roman Vershynin, Terence Tao
Suppose we wish to transmit a vector f ∈ Rn reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are...
Rejoinder: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$ (2008)
Candès, Emmanuel, Tao, Terence
Rejoinder to ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]
Why are solitons stable? (2008)
The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once...
A positive proof of the Littlewood-Richardson rule using the octahedron recurrence (2008)
Allen Knutson, Terence Tao, Christopher Woodward
the octahedron recurrence
Finite bounds for Hölder–Brascamp– Lieb multilinear inequalities, preprint (2008)
Jonathan Bennett, Anthony Carbery, Michael Christ, Terence Tao, Of Hölder
Abstract. A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties
Ill-posedness for nonlinear Schrodinger and wave equations, to appear Annales IHP (2008)
Michael Christ, James Colliander, Terence Tao
Abstract. The nonlinear wave and Schrödinger equations on R d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H s...
POINTWISE CONVERGENCE OF LACUNARY SPHERICAL MEANS (2007)
Andreas Seeger, Terence Tao, James Wright
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...
The quantitative behaviour of polynomial orbits on nilmanifolds (2007)
A theorem of Leibman asserts that a polynomial orbit (g(1),g(2),g(3),...) on a nilmanifold G/L is always equidistributed in a union of closed sub-nilmanifolds of G/L. In this paper we give a...
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007)
Killip, Rowan, Tao, Terence, Visan, Monica
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2)...
A quantitative version of the Besicovitch projection theorem via multiscale analysis (2007)
By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has...
A NOTE ON THE FREIMAN AND BALOG-SZEMERÉDI-GOWERS THEOREMS IN FINITE FIELDS (2007)
Abstract. We obtain quantitative versions of the Balog-Szemerédi-Gowersand Freiman theorems in the model case of a finite field geometry Fn 2, improving the previously known bounds in such theorems....
What is good mathematics? (2007)
We all agree that mathematicians should strive to produce good mathematics. But how does one define “good mathematics”, and should one even dare to try at all? Let us first consider the former...
What is good mathematics (2007)
Abstract. Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemerédi’s theorem...
John-type theorems for generalized arithmetic progressions and iterated sumsets (2006)
A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional)...
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? (2006)
Candès, Emmanuel J., Tao, Terence
Suppose we are given a vector f in a class F ⊂ ℝN, e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within...
A (concentration-)compact attractor for high-dimensional non-linear Schr\"odinger equations (2006)
We study the asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy...
Two remarks on the generalised Korteweg de-Vries equation (2006)
We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any...
Linear Equations in Primes (2006)
Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A...
Candès, Emmanuel J., Romberg, Justin, Tao, Terence
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl...
Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions (2006)
Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the...
Agency and the Semantic Web (2006)
Bruno De Finetti, Terence Tao, Christopher Walton, Ulf Gren, Michael Miller
p. 36
Differential and Integral Equations (2006)
Robert Aumann, Terence Tao, Peter Collins
p. 41 Mathematics Books from Oxford | www.oup.com | Oxford University Press is pleased to announce that all EMS members can now benefit from a 20 % discount on a huge range of Mathematics books.
Decoding by linear programming (2005)
Candès, Emmanuel J., Tao, Terence
This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ from corrupted measurements y=Af+e. Here, A is an m by n...
The Dantzig selector: Statistical estimation when $p$ is much larger than $n$ (2005)
Candes, Emmanuel, Tao, Terence
In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=X\beta+z$, where...
On the singularity probability of random Bernoulli matrices (2005)
Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that...
Stable Signal Recovery from Incomplete and Inaccurate Measurements (2005)
Emmanuel C, Justin Romberg, Terence Tao
Suppose we wish to recover a vector x0 ∈ R m (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax0 + e; A is a n by m matrix with far fewer rows than columns (n ≪...
Stable Signal Recovery from Incomplete and Inaccurate Measurements (2005)
Emmanuel C, Justin Romberg, Terence Tao
Suppose we wish to recover a vector x0 ∈ R m (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax0 + e; A is a n by m matrix with far fewer rows than columns (n ≪...
The Brascamp-Lieb inequalities: finiteness, structure, and extremals (2005)
Jonathan Bennett, Anthony Carbery, Michael Christ, Terence Tao
Abstract. 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...
Error Correction via Linear Programming (2005)
Suppose we wish to transmit a vector f ∈ Rn reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are...
The Dantzig selector: statistical estimation when p is much larger than n (2005)
In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y = Xβ + z, where β ∈...
An uncertainty principle for cyclic groups of prime order (2005)
Abstract. Let G be a finite abelian group, and let f: G → C be a complex function on G. The uncertainty principle asserts that the support supp(f):= {x ∈ G: f(x) �=0} is related to the support...
On random $\pm 1$ matrices: Singularity and Determinant (2004)
This papers contains two results concerning random $n \times n$ Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value $\sqrt {n!} \exp(O(\sqrt(n...
Emmanuel C, Justin Romberg, Terence Tao
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f ∈ C N and a randomly chosen set of frequencies Ω. Is it...
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? (2004)
Suppose we are given a vector f in a class F ⊂ R N, e.g. a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within...
Michael Christ, James Colliander, Terence Tao
In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrodinger (NLS), focusing modi ed Korteweg-de Vries (mKdV), and complex Korteweg-de Vries...
Singular Maximal Functions and Radon Transforms near L¹ (2002)
Andreas Seeger, Terence Tao, James Wright
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...
Carleson measures, trees, extrapolation, and T(b) theorems (2002)
Auscher, Pascal, Hofmann, Steve, Muscalu, Camil, Tao, Terence, Thiele, Christoph
The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of...
Recent progress on the Kakeya conjecture (2002)
We survey recent developments on the Kakeya problem. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
Global well-posedness for KdV in Sobolev spaces of negative index (2001)
Colliander, James, Keel, Markus, Staffilani, Gigliola, Takaoka, Hideo, Tao, Terence
The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in...
Endpoint mapping properties of spherical maximal operators, preprint (2001)
Andreas Seeger, Terence Tao, James Wright
this paper is to prove analogues of the L and L endpoint estimates for general functions in L , assuming however an additional regularity assumption (see hypothesis (R p ) below). The main general...
Global regularity of wave maps, I: small critical Sobolev norm in high dimension (2001)
We show that wave maps from Minkowski space $${\mathbf{R}}^{1+n}$$ to a sphere Sm − 1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space...
Local well-posedness of the Yang-Mills equation in the Temporal Gauge below the energy norm (2000)
We show that the Yang-Mills equation in three dimensions is locally well-posed in the Temporal gauge for initial data in H^s x H^{s-1} for s > 3/4, if the norm of the initial data is sufficiently...
In 1917 S. Kakeya posed the Kakeya needle problem: What is the smallest area required to rotate a unit line segment (a “needle”) by 180 degrees in the plane? Rotating around the midpoint requires
The honeycomb model of GLn(C) tensor products I: proof of the saturation conjecture (1999)
Abstract. Recently Klyachko [Kl] has given linear inequalities on triples (λ, µ, ν) of dominant weights of GLn(C) necessary for the the corresponding Littlewood-Richardson coefficient...
The Bochner Riesz conjecture implies the Restriction conjecture (1999)
Fix n 2. For any 1 p 1 and ff 0 we use BR(p; ff) to denote the statement that S
Low Regularity Semi-Linear Wave Equations (1999)
. We prove local well-posedness results for the semi-linear wave equation for data in H fl , 0 ! fl ! n\Gamma3 2(n\Gamma1) , extending the previously known results for this problem. The improvement...
Apiary views of the Berenstein-Zelevinsky polytope, and Klaychko's saturation conjecture (1998)
Abstract. Recently Klyachko [K] has given linear inequalities on triples (; ; ) of dominant weights of GLn (C) necessary for the the corresponding Littlewood-Richardson coefficient dim(V \Omega...
Endpoint Strichartz estimates (1998)
Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension...
Endpoint Strichartz estimates (1998)
Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrodinger equation (in dimension...
Jay X. Tang, Tadanao Ito, Terence Tao, Peter Traub, Paul A. Janmey
: A number of positively charged polypeptides and proteins bundle DNA, F-actin, microtubules, and viruses such as filamentous phage fd and tobacco mosaic virus (TMV), as well as intermediate...
Gong, Bang-Jian, Mabuchi, Katsuhide, Takahashi, Katsuhito, Nadal-Ginard, Bernardo, Tao, Terence
Calponin is a thin filament-associated protein that is implicated in the regulation and maintenance of smooth muscle contraction. Molecular cloning of chicken gizzard calponin indicated the presence...
Relative efficiencies of kernel and local likelihood density estimators
Local likelihood methods enjoy advantageous properties, such as good performance in the presence of edge effects, that are rarely found in other approaches to nonparametric density estimation....
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...
. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrodinger equation (in dimension n 3)....
Michael Christ, James Colliander, Terence Tao
Abstract. Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces H s, for s negative but close to 0. For smooth...