We consider a heat-type operator \(\mathcal {L}\) structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, with a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if \(\mathcal {L}u\) is smooth with respect to the space variables, the same is true for u, with quantitative regularity estimates in the scale of Sobolev spaces defined by right invariant vector fields. Moreover, the solution and its space derivatives of every order satisfy a 1/2-Hölder continuity estimate with respect to time. The result is proved both for weak solutions and for distributional solutions, in a suitable sense.

我们考虑一个热类型算子\（\ mathcal {L} \），该算子构造在左不变的1均匀矢量场上，该矢量场是Carnot组的生成器，其有限可测系数的正整数矩阵仅取决于时间。我们证明，如果\（\ mathcal {L} u \）相对于空间变量是平滑的，则u的情况也是如此，其中由右不变矢量场定义的Sobolev空间尺度上的定量正则性估计。此外，每个阶次的解及其空间导数都满足关于时间的1/2荷尔德连续性估计。在适当的意义上，对于弱解和分布解都证明了结果。