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A parametrix construction for wave equations with \(C^{1,1}\) coefficients. (English) Zbl 0974.35068
Summary: 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 \(n=3\).

MSC:
35L05 Wave equation
35R05 PDEs with low regular coefficients and/or low regular data
35L15 Initial value problems for second-order hyperbolic equations
35S30 Fourier integral operators applied to PDEs
35S50 Paradifferential operators as generalizations of partial differential operators in context of PDEs
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References:
[1] J.N. BONY, Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. Scient. E.N.S., 14 (1981), 209-246. · Zbl 0495.35024
[2] R.R. COIFMAN and Y. MEYER, Au delà des opérateurs pseudo-differentiels, Astérisque, Soc. Math. France, 57 (1978). · Zbl 0483.35082
[3] A. CORDOBA and C. FEFFERMAN, Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3-11 (1978), 979-1005. · Zbl 0389.35046
[4] C. FEFFERMAN, A note on spherical summation multipliers, Israel J. Math., 15 (1973), 44-52. · Zbl 0262.42007
[5] A.E. HURD and D.H. SATTINGER, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Math. Soc., 132 (1968), 159-174. · Zbl 0155.16401
[6] Y. MEYER, Ondelettes et opérateurs II, Opérateurs de Calderón-Zygmund, Hermann, Paris, 1990. · Zbl 0694.41037
[7] H. PECHER, Nonlinear small data scattering for the wave and Klein-Gordan equations, Math. Z., 185 (1984), 261-270. · Zbl 0538.35063
[8] A. SEEGER, C.D. SOGGE and E.M. STEIN, Regularity properties of Fourier integral operators, Annals Math., 133 (1991), 231-251. · Zbl 0754.58037
[9] H. SMITH, A Hardy space for Fourier integral operators, Jour. Geom. Anal., to appear. · Zbl 1031.42020
[10] H. SMITH and C. SOGGE, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., 1 (1994), 729-737. · Zbl 0832.35018
[11] H. SMITH and C. SOGGE, On the critical semilinear wave equation outside convex obstacles, Jour. Amer. Math. Soc., 8 (1995), 879-916. · Zbl 0860.35081
[12] E. M. STEIN, Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, 1993. · Zbl 0821.42001
[13] R. STRICHARTZ, A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5 (1970), 218-235. · Zbl 0189.40701
[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. · Zbl 0372.35001
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