Estimates for derivatives of the Poisson kernels on homogeneous manifolds of negative curvature
type: Item
instance of: scholarly article
title: Estimates for derivatives of the Poisson kernels on homogeneous manifolds of negative curvature
zbMATH Open document ID: 1008.22003
author: Roman Urban
published in: Mathematische Zeitschrift
publication date: 2002-12-01T00:00:00Z
review text: Let \(S\) be a rank one \(NA\) group and \({\mathcal L}=\sum_{j=1}^mY_j+Y\) a second order left-invariant operator on \(S\), where \(Y_0,Y_1,\dots,Y_m\) are linearly independent and generate the Lie algebra \(\mathfrak s\) of \(S\). This operator can be written as \[ {\mathcal L}={\mathcal L}_\gamma= \sum_j\Phi_a(X_j)^2+\Phi(X)+a^2\partial_a^2+(1-\gamma)a\partial_a, \] where \(X,X_1,X_2,\dots,X_m\) are left-invariant vector fields on \(N\), \(X_1,X_2,\dots,X_m\) are linearly independent and generate \(\mathfrak n\), \(\Phi_a\), \(a>0\), is an automorphism of \(\mathfrak n\) defined by \(e^{(\log a)D}\), \(D=ad_{Y_0}\) and \(A=\exp{\mathbb R}Y_0\). Let \(m_\gamma\), \(\gamma\geq 0\), denote the Poisson kernel of \({\mathcal L}_\gamma\). In this paper R. Urban obtains some estimates for the derivatives of \(m_\gamma\); for an index \(I=(i_1,i_2,\dots,i_k)\) and all \({\mathcal X}^I\) with \(\|{\mathcal X}^I\|\leq 1\), \[ |{\mathcal X}^Im_\gamma(x)|\leq C(1+|x|)^{-Q-\gamma-\|I\|} (\log(2+|x|))^{\|I\|_0}, \] where \({\mathcal X}^I={\mathcal X}_1^{(i_1)}\cdots{\mathcal X}_k^{(i_k)}\), \({\mathcal X}_j^{(i)}\) is an element in the symmetric tensor product of \(i\) copies of the generalized eigenspace \(V_{\lambda_j}\) of \(D\) on \(\mathfrak n\), \(Q=\sum_j\Re \lambda_j\), \(\|I\|=\sum_ji_j\Re \lambda_j\), \(\|I\|_0=\sum_ji_j(\dim V_{\lambda_j}-1)\), and \(|\cdot|\) is the homogeneous norm on \(N\). When \(\gamma>0\), this estimate has been obtained by \textit{E. Damek, A. Hulanicki} and \textit{J. Zienkiewicz} [Stud. Math. 126, 115-148 (1997; Zbl 0888.22007)] and when \(\gamma=0\), \(m_0\) is dominated by \((1+|x|)^{-Q}\) above and below by a result of \textit{E. Damek, A. Hulanicki} and \textit{R. Urban} [Rev. Mat. Iberoam. 17, 257-293 (2001; Zbl 1002.31007)]. In this paper the author uses both analytic and probabilistic techniques used in these papers to estimate Bessel and evolution kernels, that respectively appeared as vertical and horizontal components of the diffu…
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 1837112
zbMATH Keywords: Poisson kernel
zbMATH Keywords: NA group
zbMATH Keywords: homogeneous manifold
zbMATH Keywords: negative curvature
MaRDI profile type: Publication
full work available at URL: s002090100397
OpenAlex ID: W2058969406
DOI: 10.1007/S002090100397
description: scientific article; zbMATH DE number 1837112