The transmission property. (English) Zbl 0766.35088

Let \(X\) be a \(C^ \infty\) manifold and let \(Y\) be an open subset of \(X\) with \(C^ \infty\) boundary. Denote by \(r_ Y\) the restriction operator to \(Y\), and by \(e_ Y\) the extension by 0 on \(X\setminus Y\) of functions defined on \(Y\). A pseudodifferential operator \(P\) in \(X\) is said to have the transmission property with respect to \(Y\) if \(P_ Y u=r_ Y Pe_ Y u\) has a \(C^ \infty\) extension to \(\overline {Y}\) for all \(u\in r_ YC_ 0^ \infty(X)\). By means of local diffeomorphisms, the study of the transmission property can be reduced to the case when \(X=\mathbb{R}^ n\) and \(Y\) is a halfspace in \(\mathbb{R}^ n\). The authors obtain necessary and sufficient conditions for the transmission property to hold when \(Y\) is a halfspace in \(\mathbb{R}^ n\) and \(P\) has symbol in \(S_{\rho,\delta}^ m\) with \(0\leq\delta<\rho\leq 1\). Then the authors determine the mapping properties of \(P_ Y\) in Sobolev spaces when \(P\) has the transmission property.


35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
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