Sobolev algebras on nonunimodular Lie groups

Abstract

Let G be a noncompact connected Lie group and $$\rho $$ be the right Haar measure of G. Let $$\mathbf{{X}}=\{X_1,\dots ,X_q\}$$ be a family of left invariant vector fields which satisfy Hormander’s condition, and let $$\Delta =-\sum _{i=1}^qX_i^2$$ be the corresponding subLaplacian. For $$1\le p<\infty $$ and $$\alpha \ge 0$$ we define the Sobolev space $$\begin{aligned} L^p_{\alpha }(G)=\{f\in L^p(\rho ): \Delta ^{\alpha /2}f\in L^p(\rho )\}, \end{aligned}$$ endowed with the norm $$\begin{aligned} \Vert f\Vert _{\alpha ,p}=\Vert f\Vert _{p}+\Vert \Delta ^{\alpha /2}f\Vert _p, \end{aligned}$$ where we denote by $$\Vert f\Vert _p$$ the norm of f in $$L^p(\rho )$$ . In this paper we show that for all $$\alpha \ge 0$$ and $$p\in (1,\infty )$$ , the space $$L^{\infty }\cap L^p_{\alpha }(G)$$ is an algebra under pointwise product, that is, there exists a positive constant $$C_{\alpha , p}$$ such that for all $$f,g\in L^{\infty }\cap L^p_{\alpha }(G)$$ , $$fg\in L^{\infty }\cap L^p_{\alpha }(G)$$ and $$\begin{aligned} \Vert fg\Vert _{\alpha ,p}\le C_{\alpha ,p} \big (\Vert f\Vert _{\alpha ,p}\Vert g\Vert _{\infty }+\Vert f\Vert _{\infty }\Vert g\Vert _{\alpha ,p}\big ). \end{aligned}$$ Such estimates were proved by Coulhon, Russ and 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 $$c>0$$ , $$X_J=X_{j_1}\dots X_{j_m}$$ and $$j_\ell \in \{1,\dots ,q\}$$ for $$\ell =1,\dots ,m$$ . 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.