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

TL;DR

This paper establishes comprehensive $L^p$-$L^q$ boundedness criteria for pseudo-differential operators on graded Lie groups, extending previous results and including sharp Sobolev embedding theorems.

Contribution

It provides necessary and sufficient conditions for $L^p$-$L^q$ boundedness of pseudo-differential operators on graded Lie groups, with new insights into symbol classes and potential estimates.

Findings

Necessary and sufficient $L^p$-$L^q$ boundedness conditions established.

Sufficient conditions for specific $p,q$ ranges provided.

Sharp Sobolev embedding theorem for graded Lie groups proved.

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 H\"ormander symbol classes on graded Lie groups, within the range $1<p\leq 2 \leq q<\infty$. Additionally, we present a sufficient condition for the $L^p$-$L^q$ estimates of pseudo-differential operators within the range $1<p\leq q\leq 2$ or $2\leq p\leq q<\infty$. 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.