# A parametrix construction for wave equations with ${C}^{1,1}$ coefficients

Annales de l'institut Fourier (1998)

- Volume: 48, Issue: 3, page 797-835
- ISSN: 0373-0956

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topSmith, Hart F.. "A parametrix construction for wave equations with $C^{1,1}$ coefficients." Annales de l'institut Fourier 48.3 (1998): 797-835. <http://eudml.org/doc/75304>.

@article{Smith1998,

abstract = {In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions $n=2$ and $n=3$.},

author = {Smith, Hart F.},

journal = {Annales de l'institut Fourier},

keywords = {wave equation; Strichartz estimates; paradifferential operators},

language = {eng},

number = {3},

pages = {797-835},

publisher = {Association des Annales de l'Institut Fourier},

title = {A parametrix construction for wave equations with $C^\{1,1\}$ coefficients},

url = {http://eudml.org/doc/75304},

volume = {48},

year = {1998},

}

TY - JOUR

AU - Smith, Hart F.

TI - A parametrix construction for wave equations with $C^{1,1}$ coefficients

JO - Annales de l'institut Fourier

PY - 1998

PB - Association des Annales de l'Institut Fourier

VL - 48

IS - 3

SP - 797

EP - 835

AB - In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions $n=2$ and $n=3$.

LA - eng

KW - wave equation; Strichartz estimates; paradifferential operators

UR - http://eudml.org/doc/75304

ER -

## References

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- [2] R.R. COIFMAN and Y. MEYER, Au delà des opérateurs pseudo-differentiels, Astérisque, Soc. Math. France, 57 (1978). Zbl0483.35082MR81b:47061
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- [6] Y. MEYER, Ondelettes et Opérateurs II, Opérateurs de Calderón-Zygmund, Hermann, Paris, 1990. Zbl0694.41037
- [7] H. PECHER, Nonlinear small data scattering for the wave and Klein-Gordan equations, Math. Z., 185 (1984), 261-270. Zbl0538.35063MR85h:35165
- [8] A. SEEGER, C.D. SOGGE and E.M. STEIN, Regularity properties of Fourier integral operators, Annals Math., 133 (1991), 231-251. Zbl0754.58037MR92g:35252
- [9] H. SMITH, A Hardy space for Fourier integral operators, Jour. Geom. Anal., to appear. Zbl1031.42020
- [10] H. SMITH and C. SOGGE, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., 1 (1994), 729-737. Zbl0832.35018MR95h:35156
- [11] H. SMITH and C. SOGGE, On the critical semilinear wave equation outside convex obstacles, Jour. Amer. Math. Soc., 8 (1995), 879-916. Zbl0860.35081MR95m:35128
- [12] E. M. STEIN, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. Zbl0821.42001MR95c:42002
- [13] R. STRICHARTZ, A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5 (1970), 218-235. Zbl0189.40701MR41 #2231
- [14] R. STRICHARTZ, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J., 44 (1977), 705-714. Zbl0372.35001MR58 #23577

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