  The purpose of this article is to show that the spectral projector estimates
for Laplace operators can be deduced from Logunov-Malinnikova's Propagation of
smallness estimates for harmonic functions [11, 10, 9]. The main point is to
pass from the local estimates obtained in [4] (on a compact manifold) to global
estimates. We also state classical consequences in terms of observability and
control for heat equations, which are direct consequences of these spectral
projector estimates..
