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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Pseudo-differential estimates for linear parabolic operators

Author: David Ellis
Journal: Trans. Amer. Math. Soc. 184 (1973), 355-371
MSC: Primary 35K30; Secondary 35S10
MathSciNet review: 0333458
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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}$ and $ {\mathcal{K}^{r,S}}$.

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Keywords: Pseudo-differential operator, evolution operator, Gårding type partition of unity, Sobolev space, energy inequality
Article copyright: © Copyright 1973 American Mathematical Society

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