\(L^p\)-bounds for pseudo-differential operators on graded Lie groups (Q2665579)

The analysis on homogeneous Lie groups and on other types of Lie groups has received a lot of attention with new applications and further advanced in many areas in the last years. For instance, the study of the Heisenberg group and its applications are a very active research field, which plays an important role in analysis, geometry, representation theory, and so on. There exist a lot of works on nilpotent Lie groups which by themselves appear as local model in the construction of parametrices for the Kohn-Laplacian and other differential operators. The paper under review is devoted to proving sharp \(L^p\)-estimates for pseudo-differential operators on arbitrary graded Lie groups. The main result of this paper is Theorem 1.2, which extends the classical Fefferman's sharp theorem on the \(L^p\)-boundedness of pseudo-differential operators for Hörmander classes on \(\mathbb{R}^n\) to general graded Lie groups, provides a critical order, and recovers many well-known results. Among other things, the proof of Theorem 1.2 is mainly relied on the quantization procedure developed by the third author and Fischer. The authors finally discuss some consequences of the main theorem for the boundedness of operators on Sobolev and Besov spaces, for instance, the boundedness of local versions of global Hörmander classes on local Sobolev spaces on the group, and also compare the boundedness on local Sobolev spaces on \(\mathbb{R}^n\), loss of regularity and gain of regularity.