Pseudo-differential estimates for linear parabolic operators

Abstract

In recent papers, S. Kaplan and D. Ellis have used singular integral operator theory, multilinear interpolation and forms of the classical “energy inequality” to obtain results for linear parabolic operators. For higher order linear parabolic operators the local estimates were globalized by a Gårding type partition of unity. In the present paper it is shown how the theory of pseudo-differential operators can be used to study linear parabolic operators without recourse to multilinear interpolation. We also prove that the Gårding type partition of unity is square summable in the Sobolev type spaces H S {H^S} and K r , S {\mathcal {K}^{r,S}} .