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Pointwise convergence to initial data for some evolution equations on symmetric spaces (2307.09281v1)
Published 18 Jul 2023 in math.AP, math.CA, and math.FA
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 $\mathbb{X}$.