Addendum to “Concentration and lack of observability of waves in highly heterogeneous media”, [1] C. Castro, E. Zuazua ⋆ Abstract In [1] we introduced a class of 1−d wave equations with rapidly oscillat- ing H¨older continuous coefficients for which the classical boundary observ- ability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low reg- ularity coefficients. The object of this Note is to further analyze this issue. As we will see, the argument in [1] only provides sharp counterexamples to the Strichartz estimates when the coefficient ρ belongs to L ∞ . We carefully analyze this counterexamples for H¨ older continuous coefficients. We also give a new application of our construction showing that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough. 1. Introduction In [1] we introduced a counterexample to the boundary observability property of the wave equation with a density ρ ∈ C 0,s for all 0 <s< 1. Moreover, we observed that this construction could be adapted to obtain counterexamples to the Strichartz estimates for the wave equation with H¨older continuous coefficients. However, the proof in [1] is only valid when ρ ∈ L ∞ . The aim of this addendum is twofold. In section 2 we present a complete and rigorous statement on this matter complementing [1] and, in section 3, we give a new application of our construction that allows obtaining sharp limits to the Sogge’s estimates for the eigenfunctions of second order elliptic operators. ⋆ Partially Supported by Grant BFM 2002-03345 of MCYT (Spain) and Grant HPRN-CT-2002-00284 of the European program New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulation