Differential operators of the second order and harmonic analysis (Q1372599)

In this very nice paper the author gives a review of methods from non-commutative harmonic analysis in the study of partial differential operators. For simplicity he restricts to second-order differential operators of the form \(L = - \sum_{i,j=1}^m a_{ij}X_i X_j +\) lower order terms, where the \(X_j = \sum_{k=1}^d b_{jk}\partial / \partial x_k\) are smooth vector fields on a \(d\)-dimensional smooth manifold \(M\) for all \(j \in \{ 1,\ldots,m \} \) and \(m \leq d\). Moreover, the matrix \((a_{ij})\) is supposed to be real and symmetric. The operator \(L\) is called transversally elliptic in case the matrix \((a_{ij})\) is positive, or negative, definite. The author discusses three main problems related to the operator \(L\). The first is the regularity of the operator \(L\). If the vector fields satisfy the Hörmander condition of order \(r\), i.e., the vector fields \(X_1,\ldots,X_m\), together with the commutators of order at most \(r\) span the tangent space at every point of \(M\), then every transversally elliptic operator \(L\) is hypoelliptic according to a paper of \textit{L. Hörmander} [Acta Math. 119, 147-171 (1967; Zbl 0156.10701)]. More detailed and quantitative regularity results have been obtained by \textit{L. P. Rothschild} and \textit{E. M. Stein} [Acta Math. 137, 247-320 (1976; Zbl 0346.35030)] via a lifting to a (in general non-commutative) nilpotent Lie group. This is illustrated by an example. The second problem concerns the local solvability of the operator \(L\). This problem has not been solved in general, but for classes of differential operators in which \(L\) occurs as a left invariant differential operator on a nilpotent Lie group the problem has been solved. It turns out that the local solvability also depends on the lower order terms. For second-order left invariant differential operators on nilpotent Lie groups with rank two the problem was solved by the author and \textit{F. Ricci} [Math. Ann. 304, 517-547 (1996; Zbl 0848.43008)]. If \(L\) is essentially self-adjoint on \(L_2(M)\) and \(m \in L_\infty({\mathbb{R}})\) then \(m(L)\) is a bounded operator on \(L_2(M)\). The third problem is to find conditions which ensure that the operator \(m(L)\) maps \(L_2(M) \cap L_p(M)\) into \(L_p(M)\) and extends to a bounded operator on \(L_p(M)\). In particular, does there exist a Marcinkiewicz-Mikhlin-Hörmander type theorem? For sublaplacians on stratified Lie groups such a theorem exists, but the best condition on the constants is still an open problem.