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Fefferman’s SAK principle in one dimension. (English) Zbl 0956.35141
Summary: We give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if \(a\) and \(b\) are symbols in \(S^2_{1,0}\) such that \(|a|\leq b\), then for all \(\varepsilon >0\) we have the estimate \[ \|a^wu\|^2_s\leq C_{s,\varepsilon}(\|b^wu\|^2_s +\|u\|^2_{s+\varepsilon}) \] for all \(u\) in the Schwartz space, where \(\|\|_t\) is the usual \(H_t\) norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
35R45 Partial differential inequalities and systems of partial differential inequalities
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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