Sharp $L^2\rightarrow L^q$ bounds on spectral projectors for low regularity metrics

We establish $L^2\rightarrow L^q$ mapping bounds for unit-width spectral projectors associated to elliptic operators with $C^s$ coefficients, in the case $1\le s\le 2$. Examples of Smith-Sogge \cite{SmSo} show that these bounds are best possible for $q$ less than the critical index. We also show that $L^\infty$ bounds hold with the same exponent as in the case of smooth coefficients.

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Published 1 January 2006