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Multiplier ideals and microlocalization. (English) Zbl 1062.35010

Chang, Sun-Yung Alice (ed.) et al., Lectures on partial differential equations. Proceedings of the conference, Hsinchu, Taiwan, September 2000 in honor of Louis Nirenberg’s 75th birthday. Somerville, MA: International Press (ISBN 1-57146-111-6/hbk). New Stud. Adv. Math. 2, 141-151 (2003).
The paper under consideration deals with regularity estimates in Sobolev spaces for several classes of Partial Differential Equations (PDE). The proofs are obtained by the method of ideals of multipliers originally developed in studying subellipticity of the \(\overline\partial\) problem in: J. J. Kohn [Acta Math. 142, 79–122 (1979; Zbl 0395.35069)]. The corresponding technique converts the study of a priori estimates into problems in algebra and geometry. This technique is illustrated by second-order PDE with real coefficients, and it is shown how ideals of subelliptic multipliers can be used in proving hypoellipticity even for not subelliptic operators. At the end of the paper a description of these ideals on CR manifolds is given.
For the entire collection see [Zbl 1034.35002].

MSC:

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35H10 Hypoelliptic equations
35H20 Subelliptic equations

Citations:

Zbl 0395.35069
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