  We introduce some new classes of time dependent functions whose defining
properties take into account of oscillations around singularities. We study
properties of solutions to the heat equation with coefficients in these classes
which are much more singular than those allowed under the current theory. In
the case of $L^2$ potentials and $L^2$ solutions, we give a characterization of
potentials which allow the Schr\"odinger heat equation to have a positive
solution. This provides a new result on the long running problem of identifying
potentials permitting a positive solution to the Schr\"odinger equation.
  We also establish a nearly necessary and sufficient condition on certain sign
changing potentials such that the corresponding heat kernel has Gaussian upper
and lower bound.
  Some applications to the Navier-Stokes equations are given. In particular, we
derive a new type of a priori estimate for solutions of Navier-Stokes
equations. The point is that the gap between this estimate and a sufficient
condition for all time smoothness of the solution is logarithmic.
