Riesz transform via heat kernel and harmonic functions on non-compact manifolds (Q2217528)

Let \(M\) be a complete, connected and non-compact Riemannian manifold. Denote by \(d\) the geodesic distance, by \(\mu\) the Riemannian measure, and by \(\mathcal{L}\) the non-negative Laplace-Beltrami operator on \(M\). \par The paper under review deals with the study of the Riesz transform \(\nabla\mathcal{L}^{-1/2}\) on \(M\) satisfying the volume doubling condition, with doubling index \(N\) and reverse doubling index \(n\), \(n\le N\). \par In Section 2, various versions of Poincaré inequalities are provided. \par In the next section, the Riesz transform for \(p\) less than the lower dimension is studied. \par If \(n>2\), the authors prove the equivalence of the following conditions, for \(p\in (2,n)\): \((R_p)\): the Riesz transform is \(L^p\)-bounded. \((G_p)\): the gradient of the heat semigroup is \(L^p\)-bounded. \((RH_p)\): reverse \(L^p\)-Hölder inequality for the gradient of harmonic functions. \par This characterization implies that for \(p\in (2, n)\), \((R_p)\) has an open ended property and is stable under gluing operations. \par If \(p\in (\max \{N, 2\},\infty)\), \((R_p)\), \((G_p)\) and \((RH_p)\) are equivalent as a consequence of a paper of \textit{T. Coulhon} et al. [J. Funct. Anal. 278, No. 8, Article ID 108398, 67 p. (2020; Zbl 1439.53041)]. \par In the proof of the main theorem, the author adapts a technique from \textit{P. Auscher} et al. [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 911--957 (2004; Zbl 1086.58013)] and develops a new criterion for boundedness of the Riesz transform. \par In the last section, examples where the new results can be applied are given.