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Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. (English) Zbl 0959.35125
The author first establishes the estimate \[ \|R^s_{\lambda,a}\|_{L^2\to L^2_\phi}\leq c\lambda^{-\frac{s}{2}} \] for the remainder \(R^s_{\lambda,a}=T_{\lambda}A_{\lambda}- \widetilde{a}^s_\lambda T_\lambda\) of the approximation of the product of the FBI transform \(T_\lambda\) and the pseudodifferential operator \(A_\lambda\) with the symbol \(a(x,\xi/\lambda)\) by \(\widetilde{a}^s_\lambda T_\lambda\), where \(a\in C^s_x(C^\infty_0)\), \(1<s\leq 2\) and \(\widetilde{a}^s_\lambda= a+\frac 2\lambda(\overline{\partial}a)(\partial-i\lambda\xi)\). Based on this estimate the author establishes a Strichartz estimate \[ \|D^{1-\varrho+\frac{s-2}{4}}u\|_{L^p(L^q)}\leq (1+\|g\|_{C^s})\|u\|_{H^1}+\|Pu\|_{H^{\frac{s-2}{2}}} \] for the uniformly hyperbolic operator \(P(x,D) = -\partial_ig^{ij}(x)\partial_j\) under the weak smoothness hypothesis \(g^{ij}\in C^s\), \(0<s\leq 1\), where \((\rho,p,q)\) is a Strichartz triplet. Finally the author shows that the initial value problem for the quasilinear hyperbolic equation \[ \partial_i g^{ij}(u)\partial_j u = N(u, \nabla u) \] is locally well posed in \(H^s\times H^{s-1}\) for \(s\geq \frac n2+\frac 78\) if \(n= 2\) and \(s > \frac n2+\frac 34\) if \(n\geq 3\).

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B45 A priori estimates in context of PDEs
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
35L15 Initial value problems for second-order hyperbolic equations
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