Sharp L2 → Lq bounds on spectral projectors for low regularity metrics

We establish L-2 -> L-q mapping bounds for unit-width spectral projectors associated to elliptic operators with C-s coefficients, in the case 1 <= s <= 2. Examples of Smith-Sogge [6] show that these bounds are best possible for q less than the critical index. We also show that LOO bounds hold with the same exponent as in the case of smooth coefficients.