Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups
Let \(G\) be a semisimple connected noncompact real Lie group with finite center and such that the Lie algebra \({\mathfrak g}\) has no compact factor. If \(\vartheta\) is a Cartan involution of \({\mathfrak g}\), then \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) (\(\vartheta(X)= X\) if \(X\in{\mathfrak k}\), \(\vartheta(X)= -X\) if \(X\in{\mathfrak p}\)) and \(K= \exp{\mathfrak k}\) is a maximal compact subgroup. The article is devoted to the heat kernel \(h_t(x,y)= h_t(y^{-1}x)\) corresponding to the rather general class of left-invariant sub-Laplacians \(\Delta= \sum^p_{j=1} Y^2_j+ \sum^\ell_{j=1} X^2_j\), where \(Y_1,\dots, Y_p\) is an orthonormal basis of \({\mathfrak p}\) and \(X_1,\dots, X_l\) are elements from \({\mathfrak k}\). The author proves certain global estimates concerning the asymptotical behaviour of the bilaterally averaged heat kernel \(\int_{K\times K} dk_1 dk_2 h_t(k_1 xk_2)\) (they cannot be stated here) which permits him to compare \(h_t(x)\) with the heat kernel related to the Laplace-Beltrami operator on the symmetric space \(G/K\): \[ h_t(x)\leq C\;t^{-\sigma}(1+\|x\|)^\tau \varphi_0(x) e^{-t|\rho|^2-\|x\|^2/4(t+ 2)} \] for all \(t\geq 2\), \(x\in G\), where \(\|x\|\) is the Carnot-Carathéodory metric.
scientific article; zbMATH DE number 1915839
scientific article