References & Citations
Mathematics > Classical Analysis and ODEs
Title: $L^p$-asymptotic behaviour of solutions of the fractional heat equation on Riemannian symmetric spaces of noncompact type
(Submitted on 15 Apr 2024 (v1), last revised 26 Apr 2024 (this version, v2))
Abstract: The main goal of this paper is to study the $L^p$-asymptotic behaviour of solutions to the heat equation and the fractional heat equations on a Riemannian symmetric space of non-compact type. For $\alpha \in (0,1]$, let $h_t^\alpha$ denotes the fractional heat kernel on a Riemannian symmetric space of non-compact type $X=G/K$ (with $h_t^1$ being the standard heat kernel of $X$). Suppose $f \in L^1(X)$ is a left $K$-invariant function. We show that if $p \in [1,2]$ then \begin{equation*} \lim_{t\to\infty}\|h_t^\alpha\|_p^{-1}\|f\ast h_t^\alpha-\what f(i \gamma_p \rho)h_t^\alpha\|_p=0, \end{equation*} where $\gamma_p=2/p-1$, $\rho$ is the half sum of positive roots and $\what f$ denotes the spherical Fourier transform of $f$. If $p\in (2,\infty]$ then \begin{equation*} \lim_{t\to\infty}\|h_t^\alpha\|_p^{-1}\|f\ast h_t^\alpha-\what f(0)h_t^\alpha\|_p=0. \end{equation*} In fact, we will show that the above results are valid for more general class of left $K$-invariant functions larger than the space of integrable left $K$-invariant functions on $X$. The above statements for symmetric spaces were earlier studied in the important special case $p=1=\alpha$, in \cite{Vaz-2, AE}.
Submission history
From: Jayanta Sarkar [view email][v1] Mon, 15 Apr 2024 17:57:54 GMT (24kb)
[v2] Fri, 26 Apr 2024 17:45:36 GMT (25kb)
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