A criterion for hypoellipticity of second order differential operators (Q1102439)

This paper gives some sufficient condition for a second order differential operator, with real coefficients and non negative principal part, to be hypoelliptic. These conditions are modelled to include operators like \[ P = D^ 2_ t\;+\;D^ 2_{x_ 1}\;+\;\exp (- 1/| x_ 1|^{\delta})D^ 2_{x_ 2},\;\delta >0, \] \[ P = D^ 2_ t\;+\;x^ 2_ 2 D^ 2_{x_ 1}\;+\;D^ 2_{x_ 2}\;+\;D_{x_ 3}(\exp (-1/| x_ 1|^{\beta})D_{x_ 3}, \] which are hypoelliptic respectively if and only if \(\delta <1\) and \(\beta <\).