Searcharxiv⌕ Search

arXiv subjects

Filippo De Mari

Publications and source records attributed to Filippo De Mari.

At least 19 recordsLinked to original sources
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↗
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

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

Horocyclic harmonic Bergman spaces on homogeneous trees

The main focus of this contribution is on the harmonic Bergman spaces $\mathcal{B}_α^{p}$ on the $q$-homogeneous tree $\mathfrak{X}_q$ endowed with a family of measures $σ_α$ that are constant on the horocycles tangent to a fixed boundary point and turn out to be doubling with respect to the corresponding horocyclic Gromov distance. A central role is played by the reproducing kernel Hilbert space $\mathcal{B}_α^{2}$ for which we find a natural orthonormal basis and formulae for the kernel. We also consider the atomic Hardy space and the bounded mean oscillation space. Appealing to an adaptation of Calderón-Zygmund theory and to standard boundedness results for integral operators on $L^p_α$ spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection.

math.FA↗
arXiv2021

Unitarization of the Horocyclic Radon Transform on Symmetric Spaces

We consider the Radon transform for a dual pair $(X,Ξ)$, where $X=G/K$ is a noncompact symmetric space and $Ξ$ is the space of horocycles of $X$. We address the unitarization problem that was considered (and solved in some cases) by Helgason, namely the determination of a pseudo-differential operator such that the pre-composition with the Radon transform extends to a unitary operator $\mathcal{Q}\colon L^2(X)\to L_\flat^2(Ξ)$, where $L_\flat^2(Ξ)$ is a closed subspace of $L^2(Ξ)$ which accounts for the Weyl symmetries. Furthermore, we show that the unitary extension intertwines the quasi-regular representations of $G$ on $L^2(X)$ and $L_\flat^2(Ξ)$.

math.RT↗
arXiv2021

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

math.RT↗
arXiv2020

Radon Transform: Dual Pairs and Irreducible Representations

We illustrate the general point of view developed in [SIAM J. Math. Anal., 51(6), 4356-4381] that can be described as a variation of Helgason's theory of dual $G$-homogeneous pairs $(X,Ξ)$ and which allows us to prove intertwining properties and inversion formulae of many existing Radon transforms. Here we analyze in detail one of the important aspects in the theory of dual pairs, namely the injectivity of the map label-to-manifold $ξ\to\hatξ$ and we prove that it is a necessary condition for the irreducibility of the quasi-regular representation of $G$ on $L^2(Ξ)$. We further explain how the theory in [SIAM J. Math. Anal., 51(6), 4356-4381] applies to the classical Radon and X-ray transforms in $\mathbb R^3$.

math.FA↗
arXiv2019

Cone-Adapted Shearlets and Radon Transforms

We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform.

math.FA↗
arXiv2018

Unitarization and Inversion Formulae for the Radon Transform between Dual Pairs

We consider the Radon transform associated to dual pairs $(X,Ξ)$ in the sense of Helgason, with $X=G/K$ and $Ξ=G/H$, where $G=\mathbb{R}^d\rtimes K$, $K$ is a closed subgroup of ${\rm GL}(d,\mathbb{R})$ and $H$ is a closed subgroup of $G$. Under some technical assumptions, we prove that if the quasi regular representations of $G$ acting on $L^2(X)$ and $L^2(Ξ)$ are irreducible, then the Radon transform admits a unitarization intertwining the two representations. If, in addition, the representations are square integrable, we provide an inversion formula for the Radon transform based on the voice transform associated to these representations.

math.RT↗
arXiv2018

On the Atomic Decomposition of Coorbit Spaces with Non-Integrable Kernel

This paper ist concerned with recent progress in the context of coorbit space theory. Based on a square integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decomposi- tions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this paper, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.

math.FA↗
arXiv2017

Radon transform intertwines shearlets and wavelets

We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groups and translations, and the tensor product of a standard wavelet representation with a wavelet-like representation. This yields a formula for shearlet coefficients that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives in the inversion of the Radon transform.

math.FA↗
arXiv2016

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.

math.FA↗
arXiv2015

Reproducing subgroups of $Sp(2,\mathbb{R})$. Part I: algebraic classification

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.

math.GR↗
arXiv2015

Different faces of the shearlet group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.

math.GR↗
arXiv2014

Geometric classification of semidirect products in the maximal parabolic subgroup of $\operatorname{Sp}(2,\mathbb{R})$

We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means of an essentially geometric argument. This classification has already been established without geometry, under a stricter notion of equivalence, namely conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.

math.GR↗
arXiv2014

Coorbit spaces with voice in a Fréchet space

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group $G$ that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fréchet space $\mathcal T$ of functions on $G$, which generalizes the classical choice $\mathcal T=L_w^1(G)$. Our basic example is $ \mathcal T=\bigcap_{p\in(1,+\infty)} L^p(G)$, or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrödingerlets.

math.FA↗
arXiv2012

Reproducing subgroups of Sp(2,R). Part II: admissible vectors

In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation gives rise to a reproducing formula. In all the positive cases we characterize the admissible vectors with a generalized Calderón equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.

math.RT↗