Hölder regularity of subelliptic pseudodifferential operators
The author studies the Hölder regularity of subelliptic pseudo- differential operators. He exhibits examples (Theorem 1.1) to show that in general subelliptic operators do not have good behaviour in Hölder spaces \(\Lambda^{\alpha}\) and finds it necessary to introduce the notion of a diagonal operator \[ L=\partial /\partial x_{n+1}+a(x_{n+1},x)Q,\quad x_{n+1}\in {\mathbb{R}},\quad x\in {\mathbb{R}}^ n, \] Q is a first order pseudo-differential operator in x with real symbol. Let p(x,\(\xi\)) be the principal symbol of a pseudo-differential operator P at \((x_ 0,\xi^ 0)\in \Omega \times {\mathbb{R}}^ n\setminus 0\). Recall the conditions (P) of Nirenberg and Treves, namely: Im(zp)(x,\(\xi\)) does not change sign in the bicharacteristic curve of Re(zp)(x,\(\xi\)) through \((x_ 0,\xi^ 0).\) Then a typical result of the author in dimension \(n>2\) (Theorem 4.1(a)) is: If L satisfies condition (P) then for any \(f\in \Lambda^{\alpha}_{comp}\), \(\alpha >0\), there is \(u\in \Lambda^{\alpha +\delta}_{loc}\), \(Lu=f\), and any solution \(Lu=f\) is in \(\Lambda^{\alpha +\delta}_{loc}.\) Other parts of the theorem give results under the condition (\(\Psi\)) of Nirenberg and Treves. A great deal of detailed classical analysis is involved in the study that the author makes of the relevant parametrices.
subelliptic pseudo-differential operators
scientific article; zbMATH DE number 4175535
scientific article