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Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. (English) Zbl 0990.35027
For Parts I and II see [Am. J. Math. 122, 349-376 (2000; Zbl 0959.35125) and ibid. 123, 385-423 (2001; Zbl 0988.35037)].
From the introduction: We are interested in Strichartz estimates for a second order operator of the form \(P(x,D)=\partial_i g^{ij}(x) \partial_j\), which is strongly hyperbolic with respect to time.
In Part II we have shown that the full estimates hold in all dimensions for operators with \(C^2\) coefficients, and we also obtained appropriate weaker estimates for operators with \(C^s\) coefficients for \(0<s<2\). The main goal of this article is to prove that the estimates are still true if the coefficients have two derivatives in \(L^1(L^\infty)\), and then to explore some consequences of this result.
It is assumed that the matrices \((g^{ij}(x))\), \((g^{ij} (x))^{-1}\) are uniformly bounded and of signature \((1,n)\), furthermore, the surfaces \(x_0=\)const are space-like uniformly in \(x\), i.e. that \(g^{00}> c>0\).

MSC:
35B45 A priori estimates in context of PDEs
35L10 Second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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