$L^p$-bounds for pseudo-differential operators on graded Lie groups

Abstract

In this work we obtain sharp $L^p$-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to H\"ormander on $\mathbb{R}^n$. The main result extends the classical Fefferman's sharp theorem on the $L^p$-boundedness of pseudo-differential operators for H\"ormander classes on $\mathbb{R}^n$ to general graded Lie groups, also adding the borderline $\rho=\delta$.