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

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 ≤ 2 ≤ q < ∞. Additionally, we present a sufficient condition for the L^p-L^q estimates of pseudo-differential operators within the range 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞. The proofs rely on estimates of the Riesz and Bessel potentials associated with Rockland oper-ators, 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.