The purpose of this note is to provide an elementary proof of a weak version of the square function estimate on L^p(Omega) which is often used in the context of wave or Schrodinger equations to reduce estimates for data in Sobolev spaces to spectrally localized data. In the process we obtain usefull L^p(Omega) bounds for the heat kernel and its derivatives, through a simple argument which holds irrespective of the domain.