Spectrum of the Laplacian and Riesz transform on locally symmetric spaces
Let \(M\) be a complete non-compact connected Riemannian manifold. Let \(|\omega|_{L^p(\Lambda^q)}\) be the \(L^p\) norm of a \(q\)-form \(\omega\). Let \(d\) be the usual exterior derivative. The square root \(\Delta^{1/2}\) of the Laplace-Beltrami operator is characterized by the identity \(|\Delta^{1/2}f|_{L^2(\Lambda^0)}:=|df|_{L^2(\Lambda^1)}\). One says that the Riesz transformation is bounded on \(L^p\) if there exists a constant \(c_p\) so that one has the estimate \(|df|_{L^p(\Lambda^1)}\leq c_p|\Delta^{1/2}f|_{L^p(\Lambda^0)}\) for all \(f\) with compact support on \(M\). Assume that the discrete part of the spectrum of \(\Delta\) is non-trivial and that \(M\) is a local symmetric space. The authors show that the Riesz transform is bounded on \(L^p\) for all \(p\) in some open neighborhood of \(2\).
scientific article; zbMATH DE number 5527922
scientific article