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arXiv · 2307.16094

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

Duván Cardona·Vishvesh Kumar·Michael Ruzhansky
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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 $1<p\leq 2 \leq q<\infty$. 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.

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Duván Cardona, Vishvesh Kumar, Michael Ruzhansky. 2023-07-29. $L^p$-$L^q$ boundedness of pseudo-differential operators on graded Lie groups. https://arxiv.org/abs/2307.16094

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