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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
##### Keywords:
Strichartz estimates; eigenfunction estimates
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