The Riesz transformation on conical varieties (Q1125286)

Let \(N\) be a connected Riemannian manifold of dimension \(n -1\geq 1\). This paper is concerned with the investigation of the Riesz transform \(\nabla(-\Delta)^{-1/2}\) on the cone \(C(N)= R^+\times N\). When \(N\) is compact, let \(\lambda_1\) be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on \(N\) and \[ p_0 =\text{sup}\bigl\{p>1\mid p.\bigl(\tfrac n2-\sqrt{(\tfrac{n-2}{2})^2+\lambda_1}\bigr)<n\bigr\}. \] The author shows that \(\nabla(-\Delta)^{-1/2}\) is of weak type (1,1) and type \((p,p)\) for all \(1 <p <p_0\) and that it is not of type \((p,p)\) for any \(p >p_0\). When \(N\) is a complete noncompact Riemannian manifold, the author studies the behavior of the Riesz transform on the cone \(C(N)\) in several cases. For example, if \(N =G/K\) is a Riemannian symmetric space of noncompact type then \(\nabla(-\Delta)^{-1/2}\) is bounded on \(L^p(C(N))\) for all \(1< p < n\), while if \(N= R^{n-1}\) with \(n \geq 3\), then \(\nabla(- \Delta)^{-1/2}\) is not bounded on \(L^p(C(N))\) for any \(p>n\).