Hardy-Littlewood theory on unimodular groups
The following result on the \(L^\infty\)-estimation of the heat diffusion kernel \(\varphi\) associated with a left invariant subelliptic Laplacian \(\Delta\) on a connected unimodular, nonamenable real Lie group \(G\) is proved: Let \(\lambda\) be the spectral gap of \(\Delta\) and let \(Q\) denote the radical of \(G\). (a) If \(Q\) is of polynomial growth, then \[ |\varphi(t,.)|_{\infty} = O(e^{-\lambda t} t^{-q/2 - D/2}), \] where \(q\) and \(D\) are constants arising from the structure of \(G/Q\) and from bounds to the growth function of \(Q\), respectively. (b) If \(Q\) is of exponential growth, there exists a \(c > 0\) such that \[ |\varphi(t,.)|_{\infty} = O(e^{-\lambda t - c t^{1/3}}). \]
scientific article; zbMATH DE number 812597