Pseudo-differential estimates for linear parabolic operators
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- by David Ellis PDF
- Trans. Amer. Math. Soc. 184 (1973), 355-371 Request permission
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 ${\mathcal {K}^{r,S}}$.References
- David Ellis, An energy inequality for higher order linear parabolic operators and its applications, Trans. Amer. Math. Soc. 165 (1972), 167–207. MR 298482, DOI 10.1090/S0002-9947-1972-0298482-9 L. Hörmander, Linear partial differential equations, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
- Stanley Kaplan, An analogue of Gȧrding’s inequality for parabolic operators and its applications, J. Math. Mech. 19 (1969/70), 171–187. MR 0603303, DOI 10.1512/iumj.1970.19.19017
- J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. MR 176362, DOI 10.1002/cpa.3160180121
- Hitoshi Kumano-go, Algebras of pseudo-differential operators, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 31–50. MR 291896 —, Remarks on pseudo-differential operators, J. Math. Soc. Japan 21 (1969), 413-439.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 184 (1973), 355-371
- MSC: Primary 35K30; Secondary 35S10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0333458-5
- MathSciNet review: 0333458