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On the solvability of pseudodifferential equations. (English) Zbl 0897.35082
Morimoto, M. (ed.) et al., Structure of solutions of differential equations. Proceedings of the Taniguchi symposium, Katata, Japan, June 26-30, 1995 and the RIMS symposium, Kyoto, Japan, July 3-7, 1995. Singapore: World Scientific. 183-213 (1996).
The aim of this paper is to survey the progress made during the past 40 years in our understanding of the conditions for local and semiglobal solvability of linear pseudodifferential equations in the framework of distribution theory, where the main problems are wide open. After recalling the known results in essentially the chronological order, the author analyses the Lerner’s counterexample to be able to discuss its implications. In particular, the following question is suggested: what are the conditions required in addition to condition \((\psi)\) to guarantee local solvability of \(Pu=f\) with \(u\in H_{(s-1 -\deg P)}\) for every \(f\in H_{(s)}\)? As is well known, it is sufficient to prove an a priori estimate of the form \[ \sup\bigl \| u(t)\bigr \|^2\leq c\delta \int\bigl \|\partial u/ \partial t-Q(t,x,D)u \bigr \|^2 dt,\;u\in {\mathcal C}^\infty_0((-\delta, \delta) \times\mathbb{R}^n) \tag{1} \] if \(\delta\) is sufficiently small, where \(Q(t,x,\xi)\in S^1_{1,0}\) is real valued. The author proves that the estimate (1) is true if \(Q\) has no change of sign from \(+\) to \(-\) for increasing \(t\) and if the sign is independent of \(\xi\) in the sense that \(Q(t,x,\xi) Q(t,x,\eta)\geq 0\) for all \(t,x,\xi,\eta\). The proof is based on an extension of the “sharp Gårding inequality”.
For the entire collection see [Zbl 0882.00037].

35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations