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Spectral cluster estimates for $$C^{1,1}$$ metrics. (English) Zbl 1284.35149
Summary: In this paper, we establish $$L^p$$ norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is $$C^{1,1}$$. Precisely, we show that the $$L^p$$ estimates proven by Ch. D. Sogge [J. Funct. Anal. 77, No. 1, 123–138 (1988; Zbl 0641.46011)] in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge [the author and Ch. D. Sogge, Math. Res. Lett. 1, No. 6, 729–737 (1994; Zbl 0832.35018)] that such estimates fail for $$C^{1,\alpha}$$ metrics if $$\alpha < 1$$.

##### MSC:
 35J15 Second-order elliptic equations 35L05 Wave equation 42B25 Maximal functions, Littlewood-Paley theory 47G30 Pseudodifferential operators
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