Subelliptic Wave Equations with Log-Lipschitz coefficients

Abstract: In this paper we study the Cauchy problem for the wave equations for sums of squares of left invariant vector fields on compact Lie groups and also for hypoelliptic homogeneous left-invariant differential operators on graded Lie groups (the positive Rockland operators), when the time-dependent propagation speed satisfies a Log-Lipschitz condition. We prove the well-posedness in the associated Sobolev spaces exhibiting a finite loss of regularity with respect to the initial data, which is not true when the propagation speed is a ${\rm H\ddot{o}lder}$ function. We also indicate an extension to general Hilbert spaces. In the special case of the Laplacian on $\mathbb R^n$, the results boil down to the celebrated result of Colombini-De Giorgi and Spagnolo.