Propagation of smallness and control for heat equations

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold without boundary. show that using new approach Logunov and Malinnikova (2018) allows one extend type Jerison Lebeau (1999) from localisation open sets localization arbitrary non-zero Lebesgue measure; can actually go beyond non-vanishing $d- \\delta$ ($\\delta >0$ small enough) Hausdorff measure. these allow Logunov–Malinnikova’s results Finally, apply null controllability equations controls localized positive The main novelty here is drop constant coefficient assumptions Apraiz et al. (2013, 2014) (or analyticity assumption Escauriaza (2017) Moyano (2019)) deal Lipschitz coefficients. Another important get first (non-one-dimensional) exact supported measure zero sets.