On the analytic-Gevrey wave front set with respect to iterates of a class of Hörmander’s operators near points with parabolic degeneracy

This paper follows the one (Derridj in Pure Appl Funct Anal (to appear), 2020) in which we studied the analytic-Gevrey wave front set with respect to iterates of hypoelliptic second-order operators, introduced by Hörmander, but of first kind (more precisely of degenerate elliptic kind). In the present work, considering this last hypothesis not satisfied, the situation seems less easy to handle and so, we introduce a subclass of Hörmander’s operators, which we call of Kolmogorov kind, for which we can study the singularity of $$\textrm{WF}_s(u;P)$$ (see definitions in next sections). We define the index of parabolic degeneracy at a point $$(x_0,\xi _0)$$ of such P and establish results as in the case of elliptic degeneracy, but with results which seem not optimally related to microlocal index k of parabolic degeneracy. The subclass we introduce contains the Kolmogorov operator $$P=\partial _x^2+x\partial _y+\partial _t$$ mentioned by Hörmander in Baouendi and Metivier (Am J Math 104(2):287–319, 1982) as an example, for which $$k=1$$ . But in the last section of this paper, we study more closely the case of one-order parabolic degeneracy at $$(x_0,\xi _0)$$ and establish an optimal result, for operators with $$k=1$$ . In particular, for the Kolmogorov operator, one obtains an optimal result in a conic neighborhood of any point $$(x_0,\xi _0)\in \omega \times W$$ (elliptic, degenerate elliptic, parabolic or degenerate parabolic).