The heat kernel and Hardy's theorem on symmetric spaces of noncompact type (Q698360)

Let \(G\) be a connected, noncompact semisimple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). Let \(X= G/K\) be the associated noncompact symmetric space and denote by \(\Delta\) the Laplace-Beltrami operator on \(X\). Let \(h_t\) denote the time \(t\) heat kernel of \(\Delta\); that is, the solution to \(u_t=\Delta u\), \(u(0,x)= \phi(x)\), is given by the convolution \(u= h_t*\phi\) for sufficiently nice functions \(\phi\). The authors prove that if \(f: X\to\mathbb{C}\) is measurable and both \(f\) and \(\widetilde f\) satisfy certain easily stated bounds, then \(f\) is a constant multiple of \(h_t\). Here \(\widetilde f\) denotes the Helgason-Fourier transform of \(f\). This provides a nice generalization of a theorem of Hardy that characterizes the heat kernel on \(\mathbb{R}\) by bounds on the function and its Fourier transform.