arXiv subjects

Maura Salvatori

Publications and source records attributed to Maura Salvatori.

11 recordsLinked to original sources

arXiv2024

Paley--Wiener theorems on the Siegel upper half-space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.

math.CV↗

arXiv2023

Polyharmonic potential theory on the Poincaré disk

We consider the open unit disk $\mathbb{D}$ equipped with the hyperbolic metric and the associated hyperbolic Laplacian $\mathfrak{L}$. For $λ\in \mathbb{C}$ and $n \in \mathbb{N}$, a $λ$-polyharmonic function of order $n$ is a function $f: \mathbb{D} \to \mathbb{C}$ such that $(\mathfrak{L}- λ\, I)^n f = 0$. If $n =1$, one gets $λ$-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for $λ$-polyharmonic functions. For this purpose, we first determine $n^{\text{th}}$-order $λ$-Poisson kernels. Subsequently, we introduce the $λ$-polyspherical functions and determine their asymptotics at the boundary $\partial \mathbb{D}$, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the $L^2$-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the $n^{\text{th}}$-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of $λ$-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits.

math.FA↗

arXiv2021

Ahlfors regular spaces have regular subspaces of any dimension

We characterize $Q$-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each $0 < α< Q$, an $α$-regular subspace. As an application, we give an alternative simple proof of the existence of $α$-regular subspaces of a $Q$-dimensional complete Ahlfors regular metric space $(X,ρ)$, which was proved in \cite{JJKRRS}.

math.MG↗

arXiv2020

Fractional Paley-Wiener and Bernstein spaces

We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley-Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type $a$ whose restriction to the real line belongs to the homogeneous Sobolev space $\dot{W}^{s,p}$ and we call these spaces fractional Paley-Wiener if $p=2$ and fractional Bernstein spaces if $p\in(1,\infty)$, that we denote by $PW^s_a$ and $\mathcal B^{s,p}_a$, respectively. For these spaces we provide a Paley-Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel-Pólya inequalities. We conclude by discussing a number of open questions.

math.CV↗

arXiv2019

Fractional Laplacian, homogeneous Sobolev spaces and their realizations

We study the fractional Laplacian and the homogeneous Sobolev spaces on R^d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.

math.CA↗

arXiv2016

Brownian motion on treebolic space: positive harmonic functions

Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T with vertex degree p+1 < 2, the latter seen as a one-complex. In a previous paper [arXiv:1212.6151, Rev. Mat. Iberoamericana, in print] we have explored the metric structure and isometry group of that space. Relying on the analysis on strip complexes, a family of natural Laplacians with "vertical drift" and the escape to infinity of the associated Brownian motion were considered. Here, we undertake a potential theoretic study, investigating the positive harmonic functions associated with those Laplacians. The methodological subtleties stem from the singularites of treebolic space at its bifurcation lines. We first study harmonic functions on simply connected sets with "rectangular" shape that are unions of strips. We derive a Poisson representation and obtain a solution of the Dirichlet problem on sets of that type. This provides properties of the density of the induced random walk on the collection of all bifuraction lines. Subsequently, we prove that each positive harmonic function with respect to that random walk has a unique extension which is harmonic with respect to the Laplacian on treebolic space. Finally, we derive a decomposition theorem for positive harmonic functions on the entire space that leads to a characterisation of the weak Liouville property. We determine all minimal harmonic functions in those cases where our Laplacian arises from lifting a (smooth) hyperbolic Laplacian with drift from the hyperbolic plane to treebolic space.

math.PR↗

arXiv2015

On some spaces of holomorphic functions of exponential growth on a half-plane

In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obtain a Paley-Wiener theorem for $\cal M^2_ω$, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that $\cal M^p_ω$ contains functions of order 1. Moreover, we prove that the orthogonal projection from $L^p(\cal R,dω)$ into $\cal M^p_ω$ is unbounded for $p\neq2$. Furthermore, we compare the spaces $\cal M^p_ω$ with the classical Hardy and Bergman spaces, and some other Hardy-Bergman-type spaces introduced more recently.

math.CV↗

arXiv2015

Functions of exponential growth in a half-plane, sets of uniqueness and the M"untz--Sz'asz problem for the Bergman space

We introduce and study some new spaces of holomorphic functions on the right half-plane. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the M"untz--Sz'asz problem for the Bergman space, that is, the problem to characterize the sets of complex powers that form a complete set the unweighted Bergman space of a disc. In this paper, we construct a space of holomorphic functions on the right half-plane, whose sets of uniqueness correspond exactly to the sets of powers that are a complete set in Bergman space. We show that this space is a reproducing kernel Hilbert space and we prove a Paley--Wiener type theorem among several other structural properties. Moreover, we determine a sufficient condition on a set of powers to be a set of uniqueness for this space, thus providing a sufficient condition for the solution of the M"untz--Sz'asz problem for the Bergman space.

math.CV↗

arXiv2014

Brownian motion on treebolic space: escape to infinity

Treebolic space is an analog of the Sol geometry, namely, it is the horocylic product of the hyperbolic upper half plane H and the homogeneous tree T with degree p+1 > 2, the latter seen as a one-complex. Let h be the Busemann function of T with respect to a fixed boundary point. Then for real q > 1 and integer p > 1, treebolic space HT(q,p) consists of all pairs (z=x+i y,w) in H x T with h(w) = log_{q} y. It can also be obtained by glueing together horziontal strips of H in a tree-like fashion. We explain the geometry and metric of HT and exhibit a locally compact group of isometries (a horocyclic product of affine groups) that acts with compact quotient. When q=p, that group contains the amenable Baumslag-Solitar group BS(p)$ as a co-compact lattice, while when q and p are distinct, it is amenable, but non-unimodular. HT(q,p) is a key example of a strip complex in the sense of our previous paper in Advances in Mathematics 226 (2011) 992-1055. Relying on the analysis of strip complexes developed in that paper, we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularites which treebolic space (as any strip complex) has along its bifurcation lines. In this first part, we obtain the rate of escape and a central limit theorem, and describe how Brownian motion converges to the natural geometric boundary at infinity. Forthcoming work will be dedicated to positive harmonic functions.

math.PR↗

arXiv2011

Brownian motion and Harmonic functions on Sol(p,q)

The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Sol(p,q) and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Sol(p,q), and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures -p^2 and -q^2, respectively.

math.PR↗

arXiv2009

The heat semigroup and Brownian motion on strip complexes

We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing "strips" along their natural boundaries according to a given graph structure. The most familiar example is the one dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are the closed unit interval times the real line, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential selfadjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.

math.PR↗