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On Hardy spaces on worm domains
Full text linkIn this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection
on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded
pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and
we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce
and discuss the results we have obtained so far in the setting of non-smooth worm domains
Hardy spaces and the Szego projection of the non-smooth worm domain Dβ'
No full textWe define Hardy spaces Hp(Dβ'), p∈(1, ∞), on the non-smooth worm domain Dβ'=[(z1,z2)∈C2:|Imz1-log|z2|2|0. Here Wk,p denotes the Sobolev space of order k and underlying Lp norm, p∈(1, ∞). As a consequence of the Lp boundedness of S~, we prove that Hp(Dβ')∩C(Dβ') is a dense subspace of Hp(Dβ')
Shift invariant subspaces of slice L2 functions
No full textIn this paper we characterize the closed invariant subspaces for the (*-)multiplier operator of the quaternionic space of slice L2 functions. As a consequence, we obtain the innerouter factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity
Regularity of the Szegö projection on model worm domains
No full textIn this paper, we study the regularity of the Szegö projection on Lebesgue and Sobolev spaces on the boundary of the unbounded model worm domain bD'β.We consider the Hardy spaceH2(D'β). Denoting by bD'β.the boundary ofD'β, it is classical thatcan be identified with the closed subspace of L2(D'β, dσ), denoted by H2(D'β), consisting of the boundary values of functions in H2(D'β), where P : L2(D'β, dσ) →H2(D'β) is the induced Lebesgue measure. The orthogonal Hilbert space projection Ws,p (bD'β.) is called the Szegö projection. Letdenote the Lebesgue–Sobolev space on bD'β. We prove that P, initially defined on the dense subspace Wsp(bD'β)∩ L2(D'β, dσ), extends to a bounded operatorP : Wsp(bD'β)→ Wsp(bD'β) and 1 < p < ȡEand s ≥ 0
Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains
No full textIn this paper we study the regularity of the Szegő projection on Lebesgue and Sobolev spaces on the distinguished boundary of the unbounded model worm domain Dβ. We denote by db(Dβ) the distinguished boundary of Dβ and define the corresponding Hardy space H2(Dβ). This can be identified with a closed subspace of L2(db(Dβ) , dσ) , that we denote by H2(db(Dβ)) , where dσ is the naturally induced measure on db(Dβ). The orthogonal Hilbert space projection P: L2(db(Dβ) , dσ) → H2(db(Dβ)) is called the Szegő projection on the distinguished boundary. We prove that P, initially defined on the dense subspace L2∩ Lp(db(Dβ) , dσ) extends to a bounded operator P: Lp(db(Dβ) , dσ) → Lp(db(Dβ) , dσ) if and only if 21+p<21-νβ where νβ=π2β- Furthermore, we also prove that P defines a bounded operator P: Ws,2(db(Dβ) , dσ) → Ws,2(db(Dβ) , dσ) if and only if 0≤s2 where Ws.2(db(Dβ) , dσ) denotes the Sobolev space of order s and underlying L2-norm. Finally, we prove a necessary condition for the boundedness of P on Ws,p(db(Dβ) , dσ) , p∈ (1 , ∞) , the Sobolev space of order s and underlying Lp-norm
Nonabelian Ramified Coverings and L^p-boundedness of Bergman Projections in C^2
No full textIn this work, we explore the theme of L p -boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by “nice” domains (e.g., strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples, we are able to prove L p boundedness of the Bergman projection for every p ∈ (1, ∞)
Ahlfors regular spaces have regular subspaces of any dimension
No full textWe 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 [8]
Sampling in spaces of entire functions of exponential type in Cn+1
Full text linkIn 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 bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn`1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in C n`1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–P ́olya type inequality
Holomorphic function spaces on the Hartogs triangle
Full text linkThe definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the (Formula presented.) mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc
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