Poisson kernel of a class of Gruschin type operators (Q923822)

The authors consider the following linear partial differential operator with double characteristics \[ P=D^ 2_ y+D^ 2_{x_ 1}+Y^ 2D^ 2_{x_ 2}+\alpha (x,y)D_{x_ 1}+\beta (x,y)D_{x_ 2}+\gamma (x,y)D_ y+\delta (x,y) \] on \(\Omega =\{(x_ 1,x_ 2,y)\in R^ 3\); \(y>0\}.\) Here \(\alpha\),\(\beta\),\(\gamma\),\(\delta\in {\mathcal C}^{\infty}({\bar \Omega})\) and \(\beta\) satisfies the following \[ Im \beta (x,0)=0\Rightarrow | \beta (x,0)| <1.\quad \forall x\in {\mathbb{R}}^ 2. \] Under the above conditions the authors construct the Poisson operator K \({\mathcal E}'(\partial \Omega)\to {\mathcal D}'({\bar \Omega})\cap {\mathcal C}^{\infty}(\Omega)\) enjoying the following properties. (i) PK: \({\mathcal E}'(\partial \Omega)\to {\mathcal C}^{\infty}({\bar \Omega}),\) (ii) the operator trace \(_{\partial \Omega}(K)-Id\) is regularising.