Maria Vallarino

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.

Let G be the Lie group ${\mathbb{R}}^2\rtimes {\mathbb{R}}^+$ 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 $\Delta=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_i\Delta^{-1/2}$ , $i=0,1,2$ . We first show that the atomic Hardy space H¹ 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 H¹ to H¹.

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.

This paper aims to study $$A_p$$ 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 $$L^p$$ 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 $$A_p$$ A p weight is in BMO and discuss the connection between $$A_p$$ A p weights and quasisymmetric mappings.

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.

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.

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--H\"ormander 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.