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 Hormander 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 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.

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). The last two authors are supported by FWO Senior Research Grant G011522N. The third author is also supported by EPSRC grants EP/R003025/2 and EP/V005529/1. Duvan Cardona was supported by the Research Foundation-Flanders (FWO) under the postdoctoral grant No 1204824N.