arXiv subjects

Maria Vallarino

Publications and source records attributed to Maria Vallarino.

At least 19 recordsLinked to original sources

arXiv2026

Poincar\'e inequalities on hyperbolic-type spaces

We establish global $L^p$-Poincar\'e inequalities, for $ p\in[1,\infty)$, on a class of nondoubling hyperbolic-type metric measure spaces. The proof relies on a discretisation of the space, which gives rise to a Gromov hyperbolic graph, called spiderweb, quasi-isometric to the original space. We prove global Poincar\'e inequalities for spiderwebs endowed with suitable measures and develop a general transference principle from discrete graphs to metric measure spaces. Combining these results yields global Poincar\'e inequalities under natural geometric and measure assumptions on the base space.

math.CA↗

arXiv2026

The fractional Porous Medium Equation on graphs

We study the fractional porous medium equation on connected infinite graphs with no local finiteness assumption. We introduce a notion of weak dual solution adapted to the discrete setting, and establish existence results for nonnegative initial data belonging to a weighted space defined through the fractional Green function, extending beyond the classical $\ell^1$ framework. Our approach relies on weighted estimates and on a detailed analysis of the associated fractional Green function. In the particular case of infinite trees with standard weights, we establish comparison principles and derive estimates for the fractional Green function, which lead to quantitative smoothing effects for solutions.

math.AP↗

arXiv2026

Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases

We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we establish existence, uniqueness, and regularity of mild solutions for initial data in $\ell^p$ spaces, with $1\leq p<\infty$. Our approach relies on time discretization via an implicit Euler scheme and an exhaustion technique using Dirichlet subgraphs. As a by-product, we obtain existence and uniqueness results for a related time-independent equation. Finite-time extinction and positivity for solutions under a specific forcing term are also proved.

math.AP↗

arXiv2024

Endpoint estimates and sparse domination in nonhomogeneous trees

We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees $X$ that model manifolds with possibly unbounded geometry. The natural Bergman measures on $X$ may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calderón-Zygmund theory for discrete, non-locally doubling metric spaces.

math.CA↗

arXiv2024

Hardy spaces and Riesz transforms on a Lie group of exponential growth

Let $G$ be the Lie group ${\Bbb{R}}^2\rtimes {\Bbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of $G$, and consider the Laplacian $Δ=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_iΔ^{-1/2}$, \hskip3pt $i=0,1,2$. We first show that the atomic Hardy space $H^1$ in $G$ introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from $H^1$ to $H^1$.

math.FA↗

arXiv2023

$A_p$ weights on nonhomogeneous trees equipped with measures of exponential growth

This paper aims to study $A_p$ weights in the context of a class of metric measure spaces with exponential volume growth, namely infinite trees with root at infinity equipped with the geodesic distance and flow measures. Our main result is a Muckenhoupt Theorem, which is a characterization of the weights for which a suitable Hardy--Littlewood maximal operator is bounded on the corresponding weighted $L^p$ spaces. We emphasise that this result does not require any geometric assumption on the tree or any condition on the flow measure. We also prove a reverse Hölder inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an $A_p$ weight is in BMO and discuss the connection between $A_p$ weights and quasisymmetric mappings.

math.FA↗

arXiv2023

Calderón-Zygmund theory on some Lie groups of exponential growth

Let $G = N \rtimes A$, where $N$ is a stratified Lie group and $A= \mathbb R_+$ acts on $N$ via automorphic dilations. We prove that the group $G$ has the Calderón-Zygmund property, in the sense of Hebisch and Steger, with respect to a family of flow measures and metrics. This generalizes in various directions previous works by Hebisch and Steger and Martini, Ottazzi and Vallarino, and provides a new approach in the development of Calderón-Zygmund theory in Lie groups of exponential growth. We also prove a weak type $(1,1)$ estimate for the Hardy-Littlewood maximal operator naturally arising in this setting.

math.FA↗

arXiv2023

Riesz transform and spectral multipliers for the flow Laplacian on nonhomogeneous trees

Let $T$ be a locally finite tree equipped with a flow measure $m$. Let $\mathcal L$ be the flow Laplacian on $(T,m)$. We prove that the first order Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^p(m)$ for $p\in (1,\infty)$. Moreover, we prove a sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type for $\mathcal L$. In the case where $m$ is locally doubling, we also prove corresponding weak type and Hardy space endpoint bounds. This generalises results by Hebisch and Steger for the canonical flow Laplacian on homogeneous trees to the setting of nonhomogeneous trees with arbitrary flow measures. The proofs rely on approximation and perturbation arguments, which allow one to transfer to any flow tree a number of $L^p$ bounds that hold on homogeneous trees of arbitrarily large degree and are uniform in the degree.

math.FA↗

arXiv2023

Harmonic Bergman projectors on homogeneous trees

In this paper we investigate some properties of the harmonic Bergman spaces $\mathcal A^p(σ)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq p<\infty$, and $σ$ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J.~Cohen, F.~Colonna, M.~Picardello and D.~Singman. When $p=2$ they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $L^p(σ)$ for $1<p<\infty$ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hörmander's condition.

math.CV↗

arXiv2023

Hardy-Littlewood maximal operators on trees with bounded geometry

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on the class $Υ_{a,b}$, $2\leq a\leq b$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$, depending on $a$ and $b$, where the centred maximal operator is bounded on $L^p(\mathfrak T)$ for all $\mathfrak T$ in $Υ_{a,b}$. We show that there exists a tree in $Υ_{a,b}$ for which the uncentred maximal function is bounded on $L^p$ if and only if $p=\infty$. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class $Υ_{a,b}$.

math.FA↗

arXiv2023

Poincaré inequalities on graphs

We prove local $L^p$-Poincaré inequalities, $ p\in[1,\infty]$, on quasiconvex sets in infinite graphs endowed with a family of locally doubling measures, and global $L^p$-Poincaré inequalities on connected sets for flow measures on trees. We also discuss the optimality of our results.

math.FA↗

arXiv2023

Pointwise multipliers for Triebel--Lizorkin and Besov spaces on Lie groups

On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_α$ for $p,q\in (1,\infty)$ and $α>d/p$, and those of Besov spaces $B^{p,q}_α$ for $q\in [1,\infty]$, $p>d$ and $d/p< α<1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\infty]$ and $α>d/p$.

math.FA↗

arXiv2023

Heat kernel and Riesz transform for the flow Laplacian on homogeneous trees

Let $\mathbb T_{q+1}$ denote the homogeneous tree of degree $q+1$ with the standard graph distance $d$ and the canonical flow measure $μ$. The metric measure space $(\mathbb T_{q+1},d,μ)$ is of exponential growth. Let $\mathcal{L}$ denote the flow Laplacian, which is a probabilistic Laplacian self-adjoint on $L^2(μ)$. In this note, we prove some weighted $L^1$-estimates for the heat kernel associated with $\mathcal{L}$ and its gradient. As a consequence, we show that the first order Riesz transform associated with the flow Laplacian on $\mathbb T_{q+1}$ is bounded on $L^p(μ)$, for $p \in (1,2]$ and of weak type $(1,1)$. The latter result was proved in a previous paper by Hebisch and Steger: we give a different proof that might pave the way to further generalizations.

math.FA↗

arXiv2022

Riesz transform for a flow Laplacian on homogeneous trees

We prove the $L^p$-boundedness, for $p \in (1,\infty)$, of the first order Riesz transform associated to the flow Laplacian on a homogeneous tree with the canonical flow measure. This result was previously proved to hold for $p \in (1,2]$ by Hebisch and Steger, but their approach does not extend to $p>2$ as we make clear by proving a negative endpoint result for $p = \infty$ for such operator. We also consider a class of ``horizontal Riesz transforms'' corresponding to differentiation along horocycles, which inherit all the boundedness properties of the Riesz transform associated to the flow Laplacian, but for which we are also able to prove a weak type $(1,1)$ bound for the adjoint operators, in the spirit of the work by Gaudry and Sjögren in the continuous setting. The homogeneous tree with the canonical flow measure is a model case of a measure-metric space which is nondoubling, of exponential growth, does not satisfy the Cheeger isoperimetric inequality, and where the Laplacian does not have spectral gap.

math.FA↗

arXiv2022

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds $M$ with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if $\mathscr{L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H^1_\mathscr{R}(M)$ is the isomorphic image of the Goldberg type space $\mathfrak{h}^1(M)$ via the map $\mathscr{L}^{1/2} (\mathscr{I} + \mathscr{L})^{-1/2}$, a fact that is false in $\mathbb{R}^n$. Specifically, $H^1_\mathscr{R}(M)$ agrees with the Hardy type space $\mathfrak{X}^{1/2}(M)$ recently introduced by the the first three authors; as a consequence, we prove that $\mathfrak{h}^1(M)$ does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek-Ricci space $S$. Our second main result states that $H^1_\mathscr{R}(S)$, the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.

math.FA↗

arXiv2022

Schrödinger equation on noncompact symmetric spaces

We establish sharp-in-time kernel and dispersive estimates for the Schrödinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schrödinger group to the wave propagator and an improved Hadamard parametrix.

math.AP↗

arXiv2021

The Sobolev embedding constant on Lie groups

In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sobolev embedding constant on $\mathbb{R}^d$. As an application, we prove local and global Moser--Trudinger inequalities.

math.FA↗

arXiv2021

Analysis on trees with nondoubling flow measures

We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderon-Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.

math.FA↗