# Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations

@article{Tao2000MultilinearWC,
title={Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations},
author={Terence Tao},
journal={American Journal of Mathematics},
year={2000},
volume={123},
pages={839 - 908}
}
• T. Tao
• Published 29 April 2000
• Mathematics
• American Journal of Mathematics
<abstract abstract-type="TeX"><p>The <i>X<sup>s,b</sup></i> spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behavior of nonlinear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the <i>L</i><sup>2</sup> norms of the…
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## References

SHOWING 1-10 OF 71 REFERENCES
The Cauchy Problem for Higher-Order KP Equations
• Mathematics
• 1999
Abstract We study the local well-posedness of higher-order KP equations. Our well-posedness results make an essential use of a global smoothing effect for the linearized equation established in
A bilinear estimate with applications to the KdV equation
• Mathematics
• 1996
u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence
Bilinear estimates and applications to 2d NLS
• Mathematics
• 2001
The three bilinearities uv, uv, uv for functions u, v : R2×[0, T ] 7−→ C are sharply estimated in function spaces Xs,b associated to the Schrodinger operator i∂t+∆. These bilinear estimates imply
Multidimensional van der Corput and sublevel set estimates
• Mathematics
• 1999
If a function has a large derivative, then it changes rapidly, and so spends little time near any particular value. This paper is devoted to quantifying that principle for functions of several
A sharp bilinear cone restriction estimate
The purpose of this paper is to prove an essentially sharp L2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature. Fix d ≥ 3, denote variables
On the Cauchy Problem for the Zakharov System
• Mathematics
• 1997
Abstract We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data ( u (0), n (0), ∂ t n (0))∈ H k ⊕ H lscr; ⊕ H l−1 , in
A bilinear approach to cone multipliers II. Applications
• Mathematics
• 2000
Abstract. This paper is a continuation of [TV], in which new bilinear estimates for surfaces in ${\bold R}^3$ were proven. We give a concrete improvement to the square function estimate of
Finite energy solutions of the Yang-Mills equations in $\mathbb{R}^{3+1}$
• Mathematics
• 1995
Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the
Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data
• Mathematics
• 1998
We prove local and global existence from large, rough initial data for a wave map between 1+1 dimensional Minkowski space and an analytic manifold. Included here is global existence for large data in
Almost global existence for solution of semilinear klein-gordon equations with small weakly decaying cauchy data
• Mathematics
• 2000
Let T∈ be the time of exisstence of a semilinear Klein-Gordon equation with small,smooth,Cauchy data of size ∈ in space dimension d≥2 If the Cauchy data are decaying rapidly enough at infinity,and