The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip ${\displaystyle X={ℝ}^n\times ]0,a[}$. We prove a representation theorem for an arbitrary ${\displaystyle C^{2,1}}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with ${\displaystyle C^{\infty }}$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.