arXiv subjects
Publications and source records attributed to Mattia Calzi.
We prove vector-valued boundedness of (suitable) Calderon-Zygmund operators and of the (truncated) Hardy-Littlewood maximal function on a connected locally doubling metric measure space.
Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $\mu_\alpha$, $\alpha\in \mathbb T$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb C$. We show that $\mu_\alpha=c|\nabla \varphi|^{-1}\chi_{\mathrm b D \cap \varphi^{-1}(\alpha)}\cdot \mathcal H^{m-1}$, where $m$ is the dimension of the Shilov boundary $\mathrm b D$ of $D$ and $c$ is a suitable constant. Denoting with $H^2(\mu_\alpha)$ the closure of the space of holomorphic polynomials in $L^2(\mu_\alpha)$, we characterize the $\alpha$ for which $H^2(\mu_\alpha)=L^2(\mu_\alpha)$ when $D$ is a polydisc; we also provide some necessary and some sufficient conditions for general domains.
In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.
Given a symmetric Siegel domain $\mathscr D$ and a positive plurihamonic function $f$ on $\mathscr D$, we study the largest positive Radon measure $μ$ on the Silov boundary $\mathrm b \mathscr D$ of $\mathscr D$ whose Poisson integral $\mathscr P μ$ is $\leq f$. If $\mathscr D$ has no tubular irreducible factors of rank $\geq 2$, we show that $\mathscr P μ$ is plurihamonic, and that $f-\mathscr P μ$ is linear. As an application, we describe a possible analogue of the family of Clark measures associated with a holomorphic function from $\mathscr D$ into the unit disc in $\mathbb C$.
Given a bounded symmetric domain $D$, we study (positive) pluriharmonic functions on $D$ and investigate a possible analogue of the family of Clark measures associated with a holomorphic function from $D$ into the unit disc in $\mathbb C$.
In this note we present a version of Hardy's inequality on a measure space $(X,μ)$ endowed with a measurable function $N\colon X\to \mathbb R$ which replaces the absolute value on $\mathbb R$ or $\mathbb R^n$, and, more generally, the distance function from a given point when $X$ is a metric space.
We consider mixed normed Bergman spaces on homogeneous Siegel domains. In the literature, two different approaches have been considered and several results seem difficult to be compared. In this paper we compare the results available in the literature and complete the existing ones in one of the two settings. The results we present are: natural inclusions, density, completeness, reproducing properties, sampling, atomic decomposition, duality, continuity of Bergman projectors, boundary values, transference.
We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results from the more particular cases of the upper half-plane, Siegel domains of tube type over irreducible cones, or symmetric, irreducible Siegel domains of type II.
In this paper we consider the (ray) representations of the group $\mathrm{Aut}$ of biholomorphisms of the Siegel upper half-space $\mathcal U$ defined by $U_s(φ) f=(f\circ φ^{-1}) (J φ^{-1})^{s/2}$, $s\in\mathbb R$, and characterize the semi-Hilbert spaces $H$ of holomorphic functions on $\mathcal U$ satisfying the following assumptions: (a) $H$ is strongly decent; (b) $U_s$ induces a bounded ray representation of the group $\mathrm{Aff}$ of affine automorphisms of $\mathcal U$ in $H$. We use this description to improve the known characterization of the semi-Hilbert spaces of holomorphic functions on $\mathcal U$ satisfying (a) and (b) with $\mathrm{Aff}$ replaced by $\mathrm{Aut}$. In addition, we characterize the mean-periodic holomorphic functions on $\mathcal U$ under the representation $U_0$ of $\mathrm{Aff}$.
We present some old and new results on a class of invariant spaces of holomorphic functions on symmetric domains, both in their circular bounded realizations and in their unbounded realizations as Siegel domains of type II. These spaces include: weighted Bergman spaces; the Hardy space $H^2$; the Dirichlet space; holomorphic Besov spaces; the Bloch space. Our main focus will be on invariant Hilbert and semi-Hilbert spaces, but we shall also discuss minimal and maximal spaces in suitable classes of invariant Banach and semi-Banach spaces.
In this paper we consider a symmetric Siegel domain $D$ and some natural representations of the M\"obius group $G$ of its biholomorphisms and of the group $\mathrm{Aff}$ of its affine biholomorphisms. We provide a complete classification of the affinely-invariant semi-Hilbert spaces (satisfying some natural additional assumptions) on tube domains, and improve the classification of M\"obius-invariant Semi-Hilbert spaces on general domains.
We continue the study of the space $BV^α(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{α,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $α\in(0,1)$, considered in the previous works arXiv:1809.08575 and arXiv:1910.13419. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{α,p}(\mathbb R^n)=L^{α,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{α,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^α$ strongly converges to the Riesz transform as $α\to0^+$ for $H^1\cap W^{α,1}$ and $S^{α,p}$ functions. We also study the convergence of the $L^1$-norm of the $α$-rescaled fractional gradient of $W^{α,1}$ functions. To achieve the strong limiting behavior of $\nabla^α$ as $α\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.
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.
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.
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 $\mathcal{N}$.
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.
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 $\mathcal{M}$.
Given a quadratic CR manifold $\mathcal{M}$ embedded in a complex space, we study Paley-Wiener-Schwartz theorems for spaces of Schwartz functions and tempered distributions on $\mathcal{M}$.