Searcharxiv⌕ Search

arXiv subjects

Mattia Calzi

Publications and source records attributed to Mattia Calzi.

32 records · Page 2Linked to original sources
arXiv2022

Boundedness of Bergman projectors on homogeneous Siegel domains

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results moslty known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the "positive" Bergman projectors, the integral operator in which the Bergman kernel is replaced by its absolute value. We provide a useful characterization which was previously known for tube domains.

math.CV↗
arXiv2022

Carleson and Sampling measures on Bernstein spaces on Siegel CR Manifolds

In this paper we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are $L^p$-integrable with respect to a natural measure. For these spaces, we prove necessary and sufficients conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.

math.CV↗
arXiv2022

Besov Spaces of Analytic Type: Interpolation, Convolution, Fourier Multipliers, Inclusions

We consider a family of Besov spaces of analytic type on the Shilov boundary $\mathcal{N}$ of a homogeneous Siegel domain $D$, and study their properties in relation to convolution, Fourier multipliers, and complex interpolation. In addition, we study how these Besov spaces of analytic type can be compared with the `classical' Besov spaces $\mathcal{N}$.

math.FA↗
arXiv2021

Bernstein Spaces on Siegel CR Manifolds

In this paper we introduce and study Bernstein spaces on a class of quadratic CR manifolds, that we call Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to a given Siegel CR submanifold are $L^p$-integrable with respect to a natural measure. For these spaces, among other results, we prove the Plancherel-Pólya inequality, a Bernstein inequality and a sufficient condition for a sequence to be sampling.

math.CV↗
arXiv2021

A Note on Hardy Spaces on Quadratic CR Manifolds

Given a quadratic CR manifold $\mathcal{M}$ embedded in a complex space, and a holomorphic function $f$ on a tubular neighbourhood of $\mathcal{M}$, we show that the $L^p$-norms of the restriction of $f$ to the translates of $\mathcal{M}$ is decreasing for the ordering induced by the closed convex envelope of the image of the Levi form of $\mathcal{M}$.

math.CV↗
arXiv2021

Schrödinger operators on Lie groups with purely discrete spectrum

On a Lie group $G$, we investigate the discreteness of the spectrum of Schrödinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable function which is bounded from below. We prove general necessary and sufficient conditions for arbitrary potentials, and we obtain explicit characterizations when $V$ is a polynomial on $G$ or belongs to a local Muckenhoupt class. We finally discuss how to transfer our results to weighted sub-Laplacians on $G$.

math.FA↗
arXiv2021

Toeplitz and Cesàro-Type Operators on Homogeneous Siegel Domains

In this paper we study Toeplitz and Cesàro-type operators on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions for these operators to be continuous or compact, or to belong to suitable Schatten classes.

math.CV↗
arXiv2021

Carleson and reverse Carleson measures on homogeneous Siegel domains

In this paper we study Carleson and reverse Carleson measures on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions in order for a measure $μ$ to be Carleson and reverse Carleson on mixed-normed weighted Bergman spaces.

math.CV↗
arXiv2020

Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $\mathcal{L}$ on a connected and simply connected nilpotent Lie group $G$ with the aid of the corresponding \emph{Rockland} operator $\mathcal{L}_0$ on the `local' contraction $G_0$ of $G$, as well as of the corresponding Rockland operator $\mathcal{L}_\infty$ on the `global' contraction $G_\infty$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\infty$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying Mihlin conditions of every order. We also prove some Mihlin-Hörmander multiplier theorems for $\mathcal{L}$ which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherel measure' associated with $\mathcal{L}$ from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

math.FA↗
arXiv2019

Spectral Multipliers on 2-step Stratified Groups, I

Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L}, -i \mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i \mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L},-i\mathcal{T})$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

math.FA↗
arXiv2019

Spectral Multipliers on $2$-Step Stratified Groups, II

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

math.FA↗
arXiv2017

Asymptotics for the Heat Kernel on H-Type Groups

We give sharp asymptotic estimates at infinity of all radial partial derivatives of the heat kernel on H-type groups. As an application, we give a new proof of the discreteness of the spectrum of some natural sub-Riemannian Ornstein-Uhlenbeck operators on these groups.

math.CA↗
arXiv2016

Weighted sub-Laplacians on Métivier Groups: Essential Self-Adjointness and Spectrum

Let $G$ be a Métivier group and let $N$ be any homogeneous norm on $G$. For $α>0$ denote by $w_α$ the function $e^{-N^α}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_α}$ associated with the Dirichlet form $ϕ\mapsto \int_{G} |\nabla_\mathcal{H}ϕ(y)|^2 w_α(y)\, dy$, where $\nabla_\mathcal{H}$ is the horizontal gradient on $G$. Consider $\mathcal{L}^{w_α}$ with domain $C_c^\infty$. We prove that $\mathcal{L}^{w_α}$ is essentially self-adjoint when $α\geq 1$. For a particular $N$, which is the norm appearing in $\mathcal{L}$'s fundamental solution when $G$ is an H-type group, we prove that $\mathcal{L}^{w_α}$ has purely discrete spectrum if and only if $α>2$, thus proving a conjecture of J. Inglis.

math.FA↗