$L^p$-$L^q$ boundedness of pseudo-differential operators on graded Lie groups

Abstract

In this paper we establish the $L^p$-$L^q$ estimates for global pseudo-differential operators on graded Lie groups. We provide both necessary and sufficient conditions for the $L^p$-$L^q$ boundedness of pseudo-differential operators associated with the global Hörmander symbol classes on graded Lie groups, within the range $1<p\le 2\le q<\infty$. Additionally, we present a sufficient condition for the $L^p$-$L^q$ estimates of pseudo-differential operators within the range $1<p\le q\le 2$ or $2\le p\le q<\infty$. The proofs rely on estimates of the Riesz and Bessel potentials associated with Rockland operators, along with previously established results on $L^p$-boundedness of global pseudo-differential operators on graded Lie groups. Notably, as a byproduct, we also establish the sharpness of the Sobolev embedding theorem for the inhomogeneous Sobolev spaces on graded Lie groups.

24 Pages