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Federico Santagati

Publications and source records attributed to Federico Santagati.

18 recordsLinked to original sources
arXiv2026

Poincaré inequalities on hyperbolic-type spaces

We establish global $L^p$-Poincaré 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 Poincaré 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 Poincaré inequalities under natural geometric and measure assumptions on the base space.

math.CA↗
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

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↗
arXiv2025

Harmonic Bergman spaces on locally finite trees

We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on $L^p$ for every $p>1$, and of weak type $(1,1)$. We also prove necessary and sufficient conditions for the $L^p$-boundedness of the extension of a class of Toeplitz-type operators.

math.FA↗
arXiv2025

Spider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spaces

In this paper we prove that if $1 τ$, and it is of weak type $(τ,τ)$, where $τ:= \log_ab$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider's webs, with ``good connectivity properties". Our result applies to trees with bounded geometry, and Cartan--Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index $τ$ is optimal in the sense that if $p<τ$, then there exists $X$ satisfying the assumptions above such that $\mathcal M$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal M$ bounded on $L^p(X)$ if and only if $p=\infty$.

math.FA↗
arXiv2024

Carleson measures on locally finite trees

We provide a characterization of Carleson measures on locally finite trees. This characterization establishes the connection between Carleson measures and the boundedness of a suitable Poisson integral between $L^p$-spaces. Additionally, when the tree has bounded degree, we investigate the relationship between Carleson measures and BMO functions defined on the boundary of the tree.

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

Triangular maximal operators on locally finite trees

We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. Córdoba and R. Fefferman. We also establish $L^p$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded.

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

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

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

Optimal heat kernel bounds and asymptotics on Damek--Ricci spaces

We give optimal bounds for the radial, space and time derivatives of arbitrary order of the heat kernel of the Laplace--Beltrami operator on Damek--Ricci spaces. In the case of symmetric spaces of rank one, these complete and actually improve conjectured estimates by Anker and Ji. We also provide asymptotics at infinity of all the radial and time derivates of the kernel. Along the way, we provide sharp bounds for all the derivatives of the Riemannian distance and obtain analogous bounds for those of the heat kernel of the distinguished Laplacian.

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

Hardy spaces on homogeneous trees with flow measures

We consider a homogeneous tree endowed with a nondoubling flow measure $μ$ of exponential growth and a probabilistic Laplacian $\mathcal{L}$ self-adjoint with respect to $μ$. We prove that the maximal characterization in terms of the heat and the Poisson semigroup of $\mathcal{L}$ and the Riesz transform characterization of the atomic Hardy space introduced in a previous work fail.

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↗
arXiv2020

Poincaré and Hardy inequalities on homogeneous trees

We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincaré inequality.

math.AP↗