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

Volume 75 / 2000

Neil Watson Annales Polonici Mathematici 75 (2000), 281-287 DOI: 10.4064/ap-75-3-281-287


Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X = ℝ^{n} × ]0,a[$. We prove a representation theorem for an arbitrary $C^{2,1}$ 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 $C^{∞}$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.


  • Neil Watson

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