Estimates for derivatives of the Poisson kernels on homogeneous manifolds of negative curvature
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 diffusion generated by \(a^{-2}{\mathcal L}_{-\gamma}\) on \(N\times{\mathbb R}^+\). However, his approach is new and different, especially, applicable simultaneously for all \(\gamma\geq 0\).
scientific article; zbMATH DE number 1837112