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On Strichartz and eigenfunction estimates for low regularity metrics. (English) Zbl 0832.35018
Examples of \(n\)-dimensional wave operators \((n\geq 3)\), including both Lipschitz and \(C^{1, \alpha}\) metrics \((0< \alpha< 1)\), for which the Strichartz estimates fail, are given. Thus, on the scale of Hölder regularity, the assumption of \(C^{1, 1}\) regularity is necessary in order to assure that the Strichartz estimate holds. The same construction also yields for \(n\)-dimensional second-order elliptic operators \((n\geq 2)\), for which certain eigenfunction estimates fail to hold.

MSC:
35B45 A priori estimates in context of PDEs
35L05 Wave equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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