Endpoint Strichartz estimates

@article{Keel1998EndpointSE,
  title={Endpoint Strichartz estimates},
  author={Markus Terence Keel and Terence Tao},
  journal={American Journal of Mathematics},
  year={1998},
  volume={120},
  pages={955 - 980}
}
  • M. Keel, T. Tao
  • Published 1 October 1998
  • Mathematics
  • American Journal of Mathematics
<abstract abstract-type="TeX"><p>We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension <i>n</i> ≥ 4) and the Schrödinger equation (in dimension <i>n</i> ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation. 

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References

SHOWING 1-10 OF 52 REFERENCES
Generalized Strichartz Inequalities for the Wave Equation
Abstract We make a synthetic exposition of the generalized Strichartz inequalities for the wave equation obtained in [6] together with the limiting cases recently obtained in [13] with as simple
On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations
Abstract We prove existence and scattering results for semilinear wave equations with low regularity data. We also determine the minimal regularity that is needed to ensure local existence and
Local smoothing of Fourier integral operators and Carleson-Sjölin estimates
The purpose of this paper is twofold. First, if Y and Z are smooth paracompact manifolds of dimensions n ~ 2 and n + 1 , respectively, we shall prove local regularity theorems for a certain class of
Smoothing properties and retarded estimates for some dispersive evolution equations
Smoothing properties, in the form of space-time integrability properties, play an important role in the study of dispersive evolution equations. A number of them follow from a combination of general
Scattering theory in the energy space for a class of non-linear wave equations
AbstractWe study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non-linear wave equation $$\square \varphi + f(\varphi ) = 0$$ in ℝn,n≧3.
Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations
Let $u(x,t)$ be the solution of the Schrodinger or wave equation with $L_2$ initial data. We provide counterexamples to plausible conjectures involving the decay in $t$ of the $\BMO$ norm of
Existence of solutions for Schrödinger evolution equations
We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u0. We provide sufficient
A restriction theorem for the Fourier transform
. In this note we will prove a (L , LP) -restriction theorem for certain submanifolds & of codimension / > 1 in an n— dimensional Euclidean space which arise as orbits under the action of a compact
Propagation of singularities and maximal functions in the plane
SummaryIn this work we mainly generalize Bourgain's circular maximal function to include variable coefficient averages. Our techniques involve a combination of Bourgain's basic ideas plus microlocal
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman
...
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