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Journal ArticleDOI

Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

01 Sep 1977-Duke Mathematical Journal (Duke University Press)-Vol. 44, Iss: 3, pp 705-714
TL;DR: In this paper, the authors give a complete solution when S is a quadratic surface given by the duality argument for the special case S {(x, y) yZ xz I} and give the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic invariant norm.
Abstract: 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 [3], and in this case the answer is known (see [l I]). Interest in Problem B was recently signalled by 1. Segal [6] who studied the special case S {(x, y) yZ xz I} and gave the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic-invariant norm. In this paper we give a complete solution when S is a quadratic surface given by
Citations
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Journal ArticleDOI
TL;DR: In this paper, an abstract Strichartz estimate for the wave equation (in dimension n ≥ 4) and for the Schrodinger equation (n ≥ 3) was proved.
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 n ≥ 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.

2,187 citations


Additional excerpts

  • ...These results extend a long line of investigation going back to a specific space-time estimate for the linear Klein-Gordon equation in [18] and the fundamental paper of Strichartz [24] drawing the connection to the restriction theorems of Tomas and Stein....

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MonographDOI
08 Jun 2006
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Abstract: Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

1,733 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the existence problem for the IVP (1.1) problem in H(R) = L(R), where R is the energy space.
Abstract: 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 on the data, for a finite time interval, whose size depends on ‖u0‖Hs . Equation in (1.1) was derived by Korteweg and de Vries [21] as a model for long wave propagating in a channel. A large amount of work has been devoted to the existence problem for the IVP (1.1). For instance, (see [9], [10]), the inverse scattering method applies to this problem, and, under appropriate decay assumptions on the data, several existence results have been established, see [5],[6],[14],[28],[33]. Another approach, inherited from hyperbolic problems, relies on the energy estimates, and, in particular shows that (1.1) is locally well posed in H(R) for s > 3/2, (see [2],[3],[12],[29],[30],[31]). Using these results and conservation laws, global (in time) well posedness in H(R), s ≥ 2 was established, (see [3],[12],[30]). Also, global in time weak solutions in the energy space H(R) were constructed in [34]. In [13] and [22] a “local smoothing” effect for solutions of (1.1) was discovered. This, combined with the conservation laws, was used in [13] and [22] to construct global in time weak solutions with data in H(R), and even in L(R). In [16], we introduced oscillatory integral techniques, to establish local well posedness of (1.1) in H(R), s > 3/4, and hence, global (in time) well posedness in H(R), s ≥ 1. (In [16] we showed how to obtain the above mentioned result by Picard iteration in an appropriate function space.) In [4] J. Bourgain introduced new function spaces, adapted to the linear operator ∂t+∂ 3 x, for which there are good “bilinear” estimates for the nonlinear term ∂x(u /2). Using these spaces, Bourgain was able to establish local well posedness of (1.1) in H(R) = L(R), and hence, by a conservation

893 citations

References
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Journal ArticleDOI

5,201 citations


"Restrictions of Fourier transforms ..." refers background or methods in this paper

  • ...Thus u L(IR" + a) when rn > 0, the usual Sobolev space of Bessel potentials (see [7]), and u L(IR" + 1) when m 0, the homogeneous Sobolev space of Riesz potentials (say for 0 < s < n/q to avoid technicalities)....

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  • ...Choose jy) (1 + [yl)<Thenf L provided ap < n 0ehas a singularity like ixl near zero and exponential decay at infinity; see [7])....

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  • ...As an immediate application of Stein’s interpolation theorem (see [7]) we have LEMMA 2....

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Journal ArticleDOI

595 citations


"Restrictions of Fourier transforms ..." refers background in this paper

  • ...Fefferman [3], and in this case the answer is known (see [l I])....

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Journal ArticleDOI
Lars Hörmander1

294 citations

Journal ArticleDOI

270 citations


"Restrictions of Fourier transforms ..." refers background or methods in this paper

  • ...fGz(x)p(x)dx= y(z)I( fs, pdlt)(t- r)z+dt so it follows from the one-dimensional analysis of (t r)%(see [4]) that...

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  • ...The computation of (z is also given in Gelfand-Shilov [4] p....

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  • ...The computation of (z is also given in Gelfand-Shilov [4] p. 290 (an alternate derivation may be given following a similar computation in 10] p. 516)....

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  • ...Finally we want to point out that none of the methods used in this paper are new; even the computations in section 2 can be found in some form in [4]....

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  • ...The computation of the Fourier transform may be found in Gelfand-Shilov [4], p....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that convolution with (1xI2)+ and related convolutions are bounded from LI to Lq for certain values of p and q, and that p is a unique choice of p which maximizes the measure of smoothing.
Abstract: We prove that convolution with (1xI2)+ and related convolutions are bounded from LI to Lq for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing il/p-l/q, in contrast with fractional integration where il/p-l/q is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev's inequality.

183 citations