$L^p$-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type

TL;DR

The paper extends Euclidean $L^p$-asymptotics for the heat equation to noncompact Riemannian symmetric spaces by showing that, for left $K$-invariant data, the long-time behavior of $f*h_t$ is governed by a $p$-dependent spherical transform constant: $\\hat f(i\\gamma_p\\rho)$ for $p\\in[1,2]$ and $\\hat f(0)$ for $p\\in(2,\\infty]$, with analogous results for fractional kernels $h_t^\\alpha$, $\\alpha\\in(0,1)$. The approach combines spherical Fourier analysis, the Herz criterion, and Kunze–Stein convolution bounds, and is complemented by a counterexample showing the $K$-invariance assumption is essential. The work further analyzes optimality in rank-one spaces and demonstrates the failure of ball-average analogues in this setting, underscoring the special role of heat kernels in the asymptotic regime. Overall, the results provide precise $L^p$-asymptotics for heat dynamics on symmetric spaces and illuminate the structure of convolution operators in this geometric context.

Abstract

For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in $L^p(X)$, $p\in [1,\infty]$, as $t\to\infty$, with the constant $M_p(f)$ depending only on $p$ and $f$. We also prove an analogous result for the fractional heat kernels $h_t^α$, $α\in (0,1)$. The above results have recently been proved for the important special cases $p=1$ and $α= 1,\frac 12$.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Proposition 3.2