zbMATH — the first resource for mathematics

Heat kernel and Green function estimates on noncompact symmetric spaces. (English) Zbl 0942.43005
Let \(X=G/K\) be a noncompact symmetric space. The primary object of study in this paper is the heat kernel \(h_t\) on \(X\), where the heat kernel is associated to the scalar Laplace-Beltrami operator and based at the identity coset. Let \({\mathfrak g}\) and \({\mathfrak k}\) represent the Lie algebras of \(G\) and \(K\), respectively, and let \({\mathfrak g}={\mathfrak k} \oplus {\mathfrak p}\) be a decomposition of the Lie algebra of \(G\) under an involutive automorphism with fixed point set \({\mathfrak k}\). Let \({\mathfrak a}\) denote a Cartan subspace of \({\mathfrak p}\) with positive Weyl chamber \({\mathfrak a}^+\). Under the assumption that, for \(H\in\overline {{\mathfrak a}^+}\), \(|H|\leq \kappa (1+t)\) for some positive constant \(\kappa\), the authors provide optimal upper and lower bounds on \(h_t (\exp H)\). These estimates involve constants that depend upon the parameter \(\kappa\). The assumption that \(|H|\leq \kappa (1+t)\) is needed for technical reasons in the proof, and the authors conjecture that their bounds hold without this assumption. Under the same technical assumption, upper and lower bounds are established for derivatives of \(h_t\), and it is conjectured that these bounds also hold without the restriction on \(H\) and \(t\).
The fourth section of the paper considers various applications of the bounds obtained for the heat kernel \(h_t\). Precise estimates, depending on the time parameter \(t\), are obtained for the \(L^p\) norm of the heat kernel for certain values of \(p\). Upper and lower bounds are provided for the Bessel-Green-Riesz kernels and the Poisson kernel. Analyzing a natural Laplacian on \(S=(\exp {\mathfrak a})N\), the authors obtain a weak type (1,1) inequality for the heat maximal operator associated to this Laplacian. In the final section, the authors obtain asymptotics at infinity for each of the kernels considered in the paper.

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
58J35 Heat and other parabolic equation methods for PDEs on manifolds
Full Text: DOI