Analytic hypo-ellipticity and propagation of regularity for a system of microdifferential equations with non-involutory characteristics
The author studies microdifferential (i.e. analytic pseudodifferential) operators of the form \(P=P_ 1P_ 2I_ m+Q;\) here \(P_ 1\) and \(P_ 2\) are scalar microdifferential operators of order \(\ell_ 1\) and \(\ell_ 2\) respectively such that the Poisson bracket \(\{\sigma(P_ 1),\sigma(P_ 2)\}\) of their principal symbols does not vanish at a point \(x^*\) of \(\sqrt{-1}T^*{\mathbb{R}}^ n-0, Q=(Q_{ij})\) is an \(m\times m\) matrix of microdifferential operators of order at most \(\ell_ 1+\ell_ 2-1\), and \(I_ m\) is the \(m\times m\) unit matrix. Suppose that \(P_ 1\) satisfies a condition (H), or equivalently that \(P_ 1\) can be transformed into the operator \(D_ 1+\sqrt{-1}x^ k\!_ 1D_ n\) near \(x^*\!_ 0=(0,\sqrt{-1}dx_ n)\) be a quantized contact transformation with a positive odd integer k; here \(D_ j=\partial /\partial x_ j.\) Suppose also that any eigenvalue of \((\sigma_{\ell -1}(Q_{ij})(x^*)/\{\sigma(P_ 1),\sigma(P_ 2)\}(x^*))\) is not a non-negative integer, where \(\sigma_{\ell -1}\) denotes the principal symbol of order \(\ell -1\). Under these conditions, it is proved that P is microlocally analytic hypo-elliptic at \(x^*\), i.e. the sheaf homomorphism \(P:C^ m\to C^ m\) is injective at \(x^*\), where C denotes the sheaf of microfunctions. For example, this result shows that a partial differential operator \((D_ 1+\sqrt{-1}x^ k\!_ 1D_ 2)(D_ 1\!^{\ell}-\sqrt{-1}x_ 1D_ 2\!^{\ell})+Q\) is locally analytic hypoelliptic at \((0,0)\in {\mathbb{R}}^ 2\) if k and \(\ell\) are positive odd integers, Q is a partial differential operator of order at most \(\ell\), and if \(\sigma_{\ell}(Q)(0,0:0,1)\) is not an integer. - To prove this result, the author uses the theory of positivity due to \textit{P. Schapira} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 14, 121-139 (1981; Zbl 0473.58022)] and a canonical form (in the complex domain) of P which was given by the author [Invent. Math. 65, 491-525 (1982; Zbl 0493.35083)].
scientific article; zbMATH DE number 3849681
scientific article