Pointwise convergence to initial data for some evolution equations on symmetric spaces

TL;DR

This paper characterizes the weights on symmetric spaces of noncompact type for which solutions to certain evolution equations converge pointwise to initial data, extending classical results with new estimates on maximal functions.

Contribution

It provides a comprehensive analysis of pointwise convergence for evolution equations on symmetric spaces, including new weighted convergence criteria and maximal function estimates.

Findings

Characterization of weights for pointwise convergence in symmetric spaces

Establishment of vector-valued weak type (1,1) estimates

L^p estimates for local Hardy--Littlewood maximal function

Abstract

Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional heat equation, and the Caffarelli--Silvestre extension problem associated with $\mathscr{L}$, and in each of these cases we characterize the weights $v$ on $\mathbb{X}$ for which the solution converges pointwise a.e. to the initial data when the latter is in $L^{p}(v)$, $1\leq p < \infty$. As a tool, we also establish vector-valued weak type $(1,1)$ and $L^{p}$ estimates ($1<p<\infty$) for the local Hardy--Littlewood maximal function on $\mathbb{X}$.