Short-time asymptotics of heat kernels of hypoelliptic Laplacians on unimodular Lie groups
Hypoelliptic heat kernel. Non-commutative harmonic analysis
This paper uses the observation that the non-commutative harmonic analysis approach proposed by \textit{A. Agrachev}, \textit{U. Boscain}, \textit{J.-P. Gauthier} and \textit{F. Rossi} in [J. Funct. Anal. 256, No. 8, 2621--2655 (2009; Zbl 1165.58012)] may be used to compute hypoelliptic heat kernel short-time asymtotics even in situations where the exact expression of the heat kernel is impossible to compute due to intractable operator equations. The main result of the paper gives such short-time asymptotics for a unimodular type I Lie group \( G \) and a left invariant hypoelliptic sub-Laplacian \( \Delta \) satisfying the following conditions. For every irreducible unitary representation \( (\lambda,\mathcal H^\lambda) \) of \( G \) let \( \hat \Delta^\lambda:=d\lambda (\Delta) \), which is a selfadjoint operator on \(\mathcal H_\lambda \). Then it is assumed that \( \mathcal H^\lambda =L^2(\Omega_\lambda) \) for some measure space \( \Omega_\lambda \), where \( \Omega_\lambda \) is a smooth variety, and \( \hat \Delta^\lambda \) can be written as a sum \( A^\lambda + B^\lambda\), both generating strongly semigroups \( (T^\lambda (t))_{t\geq 0} \) and \( (S^\lambda (t))_{t\geq 0} \). It is supposed furthermore that for every compact subset \( K\subset \Omega_\lambda \) there exists a constant \( \alpha_{K,\lambda} \), such that for every \( f\in C_c^\infty (K) \) and for every operator \( E \) which is a composition of the operators \( B^\lambda,T^\lambda(t) \) and \( S^\lambda (t) \) one has that \( \| E(f)\|_1\leq \alpha^{k}_{K,\lambda}\| f\|_1 \), where \( k \) is the number of times the operator \(B\) appears in \( E \). For each \( t\geq 0, \) \(T^\lambda(t) \) is an integral operator with kernel \( k^\lambda_t(s,r) \) and there exists an integrable function \( K^\lambda_{t/n}(s,r) \) uniformly bounded in \( s \) by an integrable function \( G^\lambda_t(s,r) \) for all \( n\geq 1 \) such that NEWLINE\[NEWLINE\begin{multlined} (t/n)[T^\lambda(t/n)^nB^\lambda+T^\lambda(t/n)^{n-1}B^\lambda T^\lambda (t/n)+\cdots T^\lambda(t/n) B^\lambda T^\lambda(t/n)^{n-1}]f(s) =\\ t\int_{\Omega_{\lambda}}K^\lambda_{t/n}(s,r)f(r)dr. \end{multlined}NEWLINE\]NEWLINE Under these conditions, the authors use Trotter's product formula and Kato's strong Trotter product formula to show that NEWLINE\[NEWLINE [e^{t \Delta^\lambda}f] (s)=\int_{\Omega_\lambda}[k^\lambda_t(s,r)+t \lim_{n\to\infty}K^\lambda_{t/n}(s,r)] f(r)dr +[\mathcal O(t^2)f] (s) NEWLINE\]NEWLINE where the symbol \(\mathcal O(t^2) \) means an operator \( D_t \) acting on \( f \), such that \( \| D_t(f)\| _1\leq M t^2 \| f \|_1 \) for a constant \( M \) and for \( t \) small enough.NEWLINENEWLINEThe authors finally show that this theorem can be applied to the hypoelliptic sub-Laplacians on the Heisenberg, Engel and Cartan groups.
scientific article; zbMATH DE number 6039128
scientific article