Function spaces for weighted subcoercive operators
We develop a theory of Sobolev, Besov and Triebel--Lizorkin spaces associated with weighted subcoercive operators on any real connected Lie group.
math.FA↗
arXiv subjects
Marco M. Peloso
Publications and source records attributed to Marco M. Peloso.
On a higher-dimensional worm domain and its geometric properties
We construct new $3$-dimensional variants of the classical Diederich-Fornaess worm domain. We show that they are smoothly bounded, pseudoconvex, and have nontrivial Nebenhülle. We also show that their Bergman projections do not preserve the Sobolev space for sufficiently large Sobolev indices.
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On the equality of De Branges-Rovnyak and Dirichlet spaces
This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{D}_μ$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces $\mathcal{D}_μ$, when $μ$ is a finite sum of atoms. This is a generalization of a previous result by Costara--Ransford \cite{costara2013}: we make no assumptions on the Pythagorean pair $(b,a)$, and we produce new examples.
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On Toeplitz operators on $H^1(\mathbb{C}^+)$
In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1(\mathbb{C}^+)$. It is well known that there are no bounded Toeplitz operators $T_{\overlineΘ}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $Θ\in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_Θ=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overlineΘ=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $Θ=e^{iτ(\cdot)}$, with $τ>0$ $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions.
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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 $\mathcal U$.
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Duality, $BMO$ and Hankel operators on Bernstein spaces
In this paper we deal with the problem of describing the dual space $(B^1_κ)^*$ of the Bernstein space $B^1_κ$, that is the space of entire functions of exponential type at most $κ>0$ whose restriction to the real line is Lebesgue integrable. We provide several characterisations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type $κ$ whose restrictions to the real line is Lebesgue integrable. We provide several characterisations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type $κ$ whose restrictions to the real line is in a suitable $BMO$-type space, or as the space of symbols $b$ for which the Hankel operatorc $H_b$ is bounded on the Paley-Wiener space $B^2_{κ/2}$. We also provide a characterisation of $(B^1_κ)^*$ as the $BMO$ space w.r.t. the Clark measure of the inner function $e^{iκz}$ on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant $1$-spaces on the torus. Furthermore, we show that the orthogonal projection $P_κ : L^2(R)\to B^2_κ$ induces a bounded operator from $L^\infty(R)$ onto $(B^1_κ)^*$. Finally, we show that $B^1_κ$ is the dual space of the suitable $VMO$-type space or as the space of symbols $b$ for which the Hankel opertor $H_b$ on the Paley-Wiener space $B^2_{k/2}$ is compact.
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Pointwise multipliers for Triebel--Lizorkin and Besov spaces on Lie groups
On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_α$ for $p,q\in (1,\infty)$ and $α>d/p$, and those of Besov spaces $B^{p,q}_α$ for $q\in [1,\infty]$, $p>d$ and $d/p< α<1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\infty]$ and $α>d/p$.
math.FA↗
On the Theory of Bergman Spaces on Homogeneous Siegel Domains
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.
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Irregularity of the Bergman projection on smooth unbounded worm domains
In this work we consider smooth unbounded worm domains $\mathcal Z_λ$ in $\mathbb C^2$ and show that the Bergman projection, densely defined on the Sobolev spaces $H^{s,p}(\mathcal Z_λ)$, $p\in(1,\infty)$, $s\ge0$, does not extend to a bounded operator $P_λ:H^{s,p}(\mathcal Z_λ)\to H^{s,p}(\mathcal Z_λ)$ when $s>0$ or $p\neq2$. The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.
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Holomorphic Function Spaces on Homogeneous Siegel Domains
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.
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Invariant Spaces of Holomorphic Functions on the Siegel Upper Half-Space
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 $\mathrm{Aff}$.
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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.
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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.
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Sampling in spaces of entire functions of exponential type in $\mathbb C^{n+1}$
In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary $b\mathcal U$ of the Siegel upper half-space $\mathcal U$ and it is fundamental that $b\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. We consider entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathcal U$ whose restriction to $b\mathcal U$ are square integrable with respect to the Haar measure on $\mathbb H_n$. For these functions we prove a version of the Whittaker--Kotelnikov--Shannon Theorem. Instrumental in our work are spaces of entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathcal U$ whose restrictions to $b\mathcal U$ belong to some homogeneous Sobolev space on $\mathbb H_n$. For these spaces, using the group Fourier transform on $\mathbb H_n$, we prove a Paley--Wiener type theorem and a Plancherel--Pólya type inequality.
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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↗
Fractional powers of higher order vector operators on bounded and unbounded domains
Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^2(Ω,\mathbb C\otimes\mathbb H)$. The operators that we consider are of the type $$ T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \ \ \ x=(x_1,\, x_2,\, x_3)\in \overlineΩ, $$ where $\overlineΩ$ is the closure of either a bounded domain $Ω$ with $C^1$ boundary, or an unbounded domain $Ω$ in $\mathbb R^3$ with a sufficiently regular boundary which satisfy the so called property $(R)$, $\{e_1,\, e_2,\, e_3\}$ is an orthonormal basis for the imaginary units of $\mathbb H$, $a_1,\,a_2,\, a_3: \overlineΩ \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In particular it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_α(T)$ for $α\in(0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial_{x_l}^m$, do not commute among themselves.
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The Sobolev embedding constant on Lie groups
In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sobolev embedding constant on $\mathbb{R}^d$. As an application, we prove local and global Moser--Trudinger inequalities.
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The Drury--Arveson space on the Siegel upper half-space and a von Neumann type inequality
In this work we study what we call Siegel--dissipative vector of commuting operators $(A_1,\ldots, A_{d+1})$ on a Hilbert space $\mathcal H$ and we obtain a von Neumann type inequality which involves the Drury--Arveson space $DA$ on the Siegel upper half-space $\mathcal U$. The operator $A_{d+1}$ is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $\{e^{-iτA_{d+1}}\}_{τ<0}$. We then study the operator $e^{-iτA_{d+1}}A^α$ where $A^α=A_1^{α_1}\cdots A^{α_d}_d$ for $α\in\mathbb N^d_0$ and prove that can be studied by means of model operators on a weighted $L^2$ space. To prove our results we obtain a Paley--Wiener type theorem for $DA$ and we investigate some multiplier operators on $DA$ as well.
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