Effie Papageorgiou

We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $$p \in [1, \infty ]$$ p ∈ [ 1 , ∞ ] , we introduce a notion of a p -mass function and prove that caloric functions with initial data belonging to certain weighted- $$\ell ^1$$ ℓ 1 spaces or to the radial $$\ell ^1$$ ℓ 1 class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat–Tits framework. We characterize the spatial concentration of heat kernels in p -norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of p , and clarify the interplay between volume growth and heat diffusion.

We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion.

Let $$\mu $$ μ be a radial compactly supported distribution on a harmonic NA group. We prove that the right convolution operator $$c_{\mu }:f \mapsto f* \mu $$ c μ : f ↦ f ∗ μ maps the space of smooth $$\mathfrak {v}$$ v -radial functions onto itself if and only if the spherical Fourier transform $$\widetilde{\mu }(\lambda )$$ μ ~ ( λ ) , $$\lambda \in \mathbb {C}$$ λ ∈ C , is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $$\mathfrak {v}$$ v -radial functions.

Let $μ$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_μ:f \mapsto f* μ$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the spherical Fourier transform $\widetildeμ(λ)$, $λ\in \mathbb{C}$, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $\mathfrak{v}$-radial functions.

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$L 1 asymptotic convergence without the assumption of bi-K-invariance.

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$L1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.