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In this paper we establish the $L^p$-$L^q$ estimates for globalpseudo-differential operators on graded Lie groups. We provide both necessaryand sufficient conditions for the $L^p$-$L^q$ boundedness ofpseudo-differential operators associated with the global H\"ormander symbolclasses on graded Lie groups, within the range $1<p\leq 2 \leq q<\infty$.Additionally, we present a sufficient condition for the $L^p$-$L^q$ estimatesof pseudo-differential operators within the range $1<p\leq q\leq 2$ or $2\leqp\leq q<\infty$. The proofs rely on estimates of the Riesz and Besselpotentials associated with Rockland operators, along with previouslyestablished results on $L^p$-boundedness of global pseudo-differentialoperators on graded Lie groups. Notably, as a byproduct, we also establish thesharpness of the Sobolev embedding theorem for the inhomogeneous Sobolev spaceson graded Lie groups.

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