arXiv subjects

Effie Papageorgiou

Publications and source records attributed to Effie Papageorgiou.

At least 19 recordsLinked to original sources

arXiv2026

Uncentred maximal operators with respect to half balls on Damek--Ricci spaces

In this paper we study a variant of the uncentred Hardy--Littlewood maximal operator on Damek--Ricci spaces in which balls are replaced by suitable half balls. Perhaps surprisingly, such modified maximal operator has better boundedness properties than the classical one. In particular, it is bounded on $L^p$ for every $p$ in $(1,\infty]$ (whereas the analogue operator on balls is bounded on $L^p$ only for $p>2$), and satisfies a limiting distributional inequality if $f$ is in $L\log (2+L)$. This endpoint estimate is optimal in the sense that it does not hold if $L\log ({2}+L)$ is replaced by a larger (in a suitable sense) Orlicz space.

math.FA↗

arXiv2026

Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $λ_{0}$, and $σ\in (0,1)$. The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u,\] by proving that nontrivial positive global solutions exist if and only if $γ\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[ Δ^σ v - λ^σ v - v^γ=0 \] for $0\leq λ\leq λ_{0}$ and $1<γ< \frac{n+2σ}{n-2σ}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former. \smallskip At the core of our results stands a novel fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties.

math.AP↗

arXiv2026

Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees

We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to\infty$. Second, using them, we show that solutions with initial data in weighted $\ell^1$ classes, asymptotically factorize in $\ell^p$ norms, $p\in[1,\infty]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion.

math.AP↗

arXiv2025

Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings

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.

math.FA↗

arXiv2024

Counting function estimates for coherent frames and Riesz sequences

We prove various estimates for the asymptotics of counting functions associated to point sets of coherent frames and Riesz sequences. The obtained results recover the necessary density conditions for coherent frames and Riesz sequences for general unimodular amenable groups, while providing more precise estimates under additional localization conditions on the coherent system for groups of polynomial growth.

math.FA↗

arXiv2024

$L^p$ asymptotics for the heat equation on symmetric spaces for non-symmetric solutions

The main goal of this work is to study the $L^p$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_0$ compactly supported or in a weighted $L^1(G/K)$ space with a weight depending on $p\in [1, \infty]$, we introduce a mass function $M_p(u_0)(\cdot)$, and prove that if $h_t$ is the heat kernel on $G/K$, then $$\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty.$$ Interestingly, the $L^p$ heat concentration leads to completely different expressions of the mass function for $1\leq p <2$ and $2\leq p\leq \infty$. If we further assume that the initial data are bi-$K$-invariant, then our mass function boils down to the constant $\int_{G/K}u_0$ in the case $p=1$, and more generally to $\mathcal{H}{u_0}(iρ(2/p-1))$ if $1\leq p<2$, and to $\mathcal{H}{u_0}(0)$ if $2\leq p \leq \infty$. Thus we improve upon results by Vázquez, Anker et al, Naik et al, clarifying the nature of the problem.

math.AP↗

arXiv2024

Surjectivity of convolution operators on harmonic $NA$ groups

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.

math.FA↗

arXiv2023

Large sets containing no copies of a given infinite sequence

Suppose $a_n$ is a real, nonnegative sequence that does not increase exponentially. For any $p<1$ we contruct a Lebesgue measurable set $E \subseteq \mathbb{R}$ which has measure at least $p$ in any unit interval and which contains no affine copy $\{x+ta_n:\ n\in\mathbb{N}\}$ of the given sequence (for any $x \in \mathbb{R}, t > 0$). We generalize this to higher dimensions and also for some ``non-linear'' copies of the sequence. Our method is probabilistic.

math.CA↗

arXiv2023

Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds

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 $L^1$ 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.

math.AP↗

arXiv2023

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

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 $L^1$ initial data. In the case of the Laplace-Beltrami operator, we show that if the initial data is 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 $L^1$ asymptotic convergence without the assumption of bi-$K$-invariance.

math.AP↗

arXiv2023

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

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 $\mathbb{X}$.

math.AP↗

arXiv2023

New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation

We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In \cite{FotDask}, harmonic maps are related to the sinh-Gordon equation and a B{ä}cklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.

math.DG↗

arXiv2023

Asymptotic behavior of the heat semigroup on certain Riemannian manifolds

We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to the previously known results in negatively curved contexts, the radiality assumption on the initial data is not required. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. Moreover, we provide a counterexample such that this asymptotic phenomenon fails in sup norm on manifolds with two Euclidean ends.

math.DG↗

arXiv2023

Asymptotics for the infinite Brownian loop on noncompact symmetric spaces

The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin when $T \rightarrow +\infty$. The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence without requiring bi-$K$-invariance for initial data, and strong $L^{\infty}$ convergence.

math.AP↗

arXiv2022

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-$K$-invariant $L^{1}$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-$K$-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-$K$-invariance condition and strong $L^{\infty}$ convergence.

math.AP↗

arXiv2022

How many Fourier coefficients are needed?

We are looking at families of functions or measures on the torus which are specified by a finite number of parameters $N$. The task, for a given family, is to look at a small number of Fourier coefficients of the object, at a set of locations that is predetermined and may depend only on $N$, and determine the object. We look at (a) the indicator functions of at most $N$ intervals of the torus and (b) at sums of at most $N$ complex point masses on the multidimensional torus. In the first case we reprove a theorem of Courtney which says that the Fourier coefficients at the locations $0, 1, \ldots, N$ are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size $O(N \log^{d-1} N)$ which suffices to determine the measure.

math.CA↗

arXiv2022

Functions tiling with several lattices

We study the problem of finding a function $f$ with ``small support'' that simultaneously tiles with finitely many lattices $Λ_1, \ldots, Λ_N$ in $d$-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.

math.MG↗

arXiv2022

Riesz means on locally symmetric spaces

We prove that for a certain class of $n$ dimensional rank one locally symmetric spaces, if $f \in L^p$, $1\leq p \leq 2$, then the Riesz means of order $z$ of $f$ converge to $f$ almost everywhere, for $\operatorname{Re}z> (n-1)(1/p-1/2).$

math.FA↗