A small example of non-local operators having no transmission property. (Q1848103)

The transmission property of a linear operator \(P\) on \(\mathbb{R}^d\) with respect to the boundary \(\mathbb{R}^{d-1}\) means that the transformation \[ {\mathcal C}^\infty_0 (\overline{\mathbb{R}^d_+})\partial u \mapsto(Pu) |_{\mathbb{R}^d} \in{\mathcal C}^\infty (\mathbb{R}^d_+),\;u:=\overline u \text{ on }\mathbb{R}^d_+,\;\overline u:=0\text{ on }\mathbb{R}^d_-, \] is continuous. The author shows that the pseudodifferential operator \(\sigma(n,D)\) having the symbol \(\sigma(n,\gamma)\), which is the composition of a general continuous negative definite function \(\psi(x,\gamma)\) of quadratic type with a complete Bernstein function \(f(\xi)\), does not have the transmission property. This result should be interesting since the generator of a Feller semigroup is a pseudodifferential operator \(\sigma (x,D)\) for which the symbol \(\sigma(x,\xi)\) is a general negative definite symbol.