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

By Terence Tao. 2000

388 citations recorded by the source corpus.

The X s,b 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 L 2 norms of the component functions. In this paper we systematically study weighted convolution estimates on L 2 . As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schrodinger X s,b spaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a7f0d34278956661579c76eaf736fdc3398d920b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R) (2008, methodology): “MR 2002k:35283 [21] T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations 232 (2007) 623-651.”
Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation (2010, background): “We firstly list some useful notations and properties for multi-linear expressions [7].”
Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr (2004, background): “In particular, our theorem implies that for any smooth data u0, there exists a unique global smooth solution.”
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces (2005, background): “Such L 2 estimates are closely connected to bilinear estimates in the spaces X σ,b (see [21] for a more general discussion).”
Sharp ill-posedness results for the KdV and mKdV equations on the torus (2011, methodology): “Finally we will make a frequent use of the following bilinear estimates that can be deduced for instance from [22] (we give a proof of these estimates in the appendix since we need to quantify the dependence of these estimates with respect to the period λ): Lemma 3.”
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, background): “13 A more detailed argument exploiting the complex conjugates as in [10, 24, 31] might obtain a better exponent in (3.”
On well-posedness for the Benjamin–Ono equation (2005, background): “The bilinear estimate may be obtained from [24] or from the following elementary argument: ∣∣∣∣∫ e−it|ξ−η|2 ḡ(η − ξ)eit|η|2f(η) dη∣∣∣∣2 = ∫ e2itξ(η−λ)f(η)f̄(λ)ḡ(η − ξ)g(λ− ξ) dη dλ ∫ ∣∣∣∣∫ e−it|ξ−η|2 ḡ(η − ξ)eit|η|2f(η) dη∣∣∣∣2 dt = ∫ δ(2ξ(η − λ))f(η)f̄(λ)ḡ(η − ξ)g(λ− ξ) dη dλ ∫ ∣∣∣(e−it|·|2 ḡ ∗ eit”
BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS (2001, background): “…point is then the following: If we have a space X s with the above properties (i.e., X s embeds in the data continuation space and (5.4) holds), and if the estimates (5.1) and (5.2) for the nonlinearity N are true, then the equation u = N (u) is locally well-posed for initial data in Hs ×Hs−1.”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, methodology): “[23] T. Tao, Global well-posedness of the Benjamin–Ono equation in H1(R), J. Hyperbolic Differ.”