## Description

**Hypoellipticity and Rockland operators**

On compact Lie groups, the Fourier analysis and the symbolic calculus developed in [RT10a] are based on the Laplacian and on the growth rate of its eigenvalues. While on compact Lie groups the Laplacians (or the Casimir element) are operators naturally associated to the group, it is no longer the case in the nilpotent setting. Thus, on nilpotent Lie groups it is natural to work with operators associated with the group through its Lie algebra structure. On stratiﬁed Lie groups these are the sub-Laplacians, and such operators are not elliptic but hypoelliptic. More generally, on graded Lie groups invariant hypoelliptic diﬀerential operators are the so-called Rockland operators.

Indeed, in [Roc78], Rockland showed that if T is a homogeneous left-invariant diﬀerential operators on the Heisenberg group, then the hypoellipticity of T and T t is equivalent to a condition now called the Rockland condition (see Deﬁnition 4.1.1). He also asked whether this equivalence would be true for more general homogeneous Lie groups. Soon after, Beals showed in [Bea77b] that the hypoel-lipticity of a homogeneous left-invariant diﬀerential operator on any homogeneous Lie group implies the Rockland condition. In the same paper he also showed that the converse holds in some step-two cases. Eventually in [HN79], Helﬀer and Nour-rigat settled what has become known as Rockland’s conjecture by proving that the hypoellipticity is equivalent to the Rockland condition (see Section 4.1.3). At the same time, it was shown by Miller [Mil80] that in the setting of homogeneous Lie groups, the existence of an operator satisfying the Rockland condition (hence of an invariant hypoelliptic diﬀerential operator in view of Helﬀer and Nourrigat’s result), implies that the group is graded, see also Section 4.1.1. This means, alto-gether, that the setting of graded Lie groups is the right generality for marrying the harmonic analysis techniques with those coming from the theory of partial diﬀerential equations.

A number of well-known functional inequalities can be extended to the graded setting, for example, see Bahouri, Fermanian-Kammerer and Gallagher [BFKG12b]. Also, there are many contributions to questions of solvability related to the hy-poellipticity problem: for a good introduction to local and non-local solvability questions on nilpotent Lie groups see Corwin and Rothschild [CR81] and, miss-ing to mention many contributions, for a more recent discussion of the topic see M¨uller, Peloso and Ricci [MPR99].

The hypoellipticity of second order operators is a very well researched sub-ject. Its beginning may be traced to the 19th century with the diﬀusion problems in probability arising in Kolmogorov’s work [Kol34]. H¨ormander made a major contribution [H¨or67b] to the subject which then developed rapidly after that (see e.g. the book of Oleinik and Radkevich [OR73]) until nowadays. We will not be concerned much with these nor with the solvability problems in this book, since one of topics of importance to us will be Rockland operators of an arbitrary degree, and we will be giving more relevant references as we go along.

Here we want to mention that the question of the analytic hypoellipticity turns out to be more involved than that in the smooth setting. In general, if a graded Lie group is not stratiﬁed, there are no homogeneous analytic hypoellip-tic left-invariant diﬀerential operators, a result by Helﬀer [Hel82]. For stratiﬁed Lie groups, the situation is roughly as follows: for H-type groups the analytic hy-poellipticity is equivalent to the smooth hypoellipticity, while for step ≥ 3 (and an additional assumption that the second stratum is one-dimensional) the sub-Laplacians are not analytic hypoelliptic, see M´etivier [M´et80] and Helﬀer [Hel82], respectively, and the discussions therein. For the Kohn-Laplacian b in the ∂¯-Neumann problem as well as for higher order operators in this setting the analytic hypoellipticity was shown earlier by Tartakoﬀ [Tar78, Tar80]. Below we will men-tion a few more facts concerning the analytic hypoellipticity in the framework of the analytic calculus of pseudo-diﬀerential operators.