Upper bound for the heat kernel on higher-rank \(NA\) groups (Q2856379)

Consider the semi-direct product \(S=N\times| A\), where \(N\) is a connected, simply connected, non-abelian nilpotent Lie group and \(A=\mathbb{R}^k\) is thought of as ordinary additive stucture. Moreover, \(N\) is considered meta-abelian. The authors assume that \(N=M\times|V\) where \(M\) and \(V\) are abelian Lie groups of dimensions \(d\) and \(n\). They define a left invariant differential operator \(\mathcal{L}_\alpha\) on \(S\) by \(\mathcal{L}_\alpha = \Delta_\alpha +\sum_{j=1}^d e^{2\xi_j(a)}Y_j^2 + \sum _{j=1}^n e^{2\theta_j(a)}X_j^2\) where \(a\in A\), \(\alpha =(\alpha_1,\dots , \alpha_k)\in \mathbb{R}^k\), \ \(\Delta_\alpha =\sum _{i=1}^k(\partial^2_{a_i}+ 2\alpha_i\partial_{a_i})\) and \(\{Y_1,\dots ,Y_d \}, \{X_1,\dots X_n\}\) are appropriate bases in the Lie algebras of \(M\) and \(V\). By using an appropriate Haar measure on \(S\) one obtains a semigroup \(T_tf(x,a)\) of operators on \(N\times|A\). This semigroup is generated by \(\mathcal{L}_\alpha\) and the kernel \(p_t\) is the heat kernel for \(\mathcal{L}_\alpha\). The authors obtain off-diagonal upper bounds for \(p_t(x,a;y,b)\). The main result of this paper is Theorem 1.1: For every \(q\geq 1\) and for every \(t\geq 0\) there is a constant \(C_{t,q}>0\) such that for every \(z\in N\) and every \(a\in A\) NEWLINE\[NEWLINEp_t(e,0;z,a)\leq C_{t,q}e^{-a^2/(32t)+\rho_0(a)}e^{-qr(z)}.NEWLINE\]NEWLINE In order to obtain this result, the authors study some homogeneous norms and a specific Riemannian metric. Then they obtain a skew-product formula and use a geometric ingredient and some moment estimates of \(C(t,y)\).