Sobolev algebras on nonunimodular Lie groups

TL;DR

This paper extends the algebra property of Sobolev spaces with pointwise multiplication from unimodular to nonunimodular Lie groups, by analyzing boundedness of local Riesz transforms and their endpoint behavior.

Contribution

It proves that Sobolev spaces on nonunimodular Lie groups form algebras under pointwise multiplication, extending previous results from unimodular groups, and studies boundedness of associated Riesz transforms.

Findings

Sobolev spaces form algebras under pointwise product on nonunimodular Lie groups.

Boundedness of local Riesz transforms on $L^p$ spaces established for large c.

Endpoint results involving Hardy and BMO spaces obtained.

Abstract

Let G be a noncompact connected Lie group and $\rho$ be the right Haar measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields which satisfy H\"ormander's condition, and let $\Delta=-\sum_{i=1}^qX_i^2$ be the corresponding subLaplacian. For $1\leq p<\infty$ and $\alpha\geq 0$ we define the Sobolev space $L^p_{\alpha}(G)={f in L^p(\rho): \Delta^{\alpha/2}f\in L^p(\rho) }$, endowed with the norm $\|f\|_{\alpha,p}=\|f\|_{p}+\|\Delta^{\alpha/2}f\|_p$, where we denote by $\|f\|_p$ the norm of $f$ in $L^p(\rho)$. In this paper we show that for all $\alpha\geq 0$ and $p\in (1,\infty)$, the space $L^{\infty}\cap L^p_{\alpha}(G)$ is an algebra under pointwise product. Such result was proved by T. Coulhon, E. Russ and V. Tardivel-Nachef in the case when G is unimodular. We shall prove it on Lie groups, thus extending their result to the nonunimodular case. In order to prove our main result, we need to study the boundedness of local Riesz transforms $R^c_J=X_J(cI+\Delta)^{-m/2}$, where c>0, $X_J=X_{j_1}...X_{j_m}$ and $j_\ell \in\{1,\dots,q\}$ for $\ell=1,...,m$. We show that if c is sufficiently large, the Riesz transform R^c_J is bounded on $L^p(\rho)$ for every $p\in (1,\infty)$, and prove also appropriate endpoint results involving Hardy and BMO spaces.