Abstract The classical Hilbert transform of a function on the real line arises as the boundary value of the function’s conjugate-harmonic extension. In this article, harmonic extensions to the upper half-integer lattice Z × N $\mathbb {Z} \times \mathbb {N}$ double struck upper Z times double struck upper N are constructed for a given boundary sequence on the integers; these serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by: (i) discrete harmonicity with respect to a two-dimensional Laplacian, (ii) a Cauchy–Riemann system, and (iii) boundary values involving a discrete Hilbert transform. This new discrete Hilbert transform is compared to the Riesz–Titchmarsh transform, and Lebesgue bounds are proved. The construction is also extended to the higher dimensional case. These results provide a discrete harmonic-analytic model analogous to the classical theory.
Abstract Let G n $G_n$ upper G Subscript n denote either the group S O ( 2 n + 1 , F ) $SO(2n+1, F)$ upper S upper O left parenthesis 2 n plus 1 comma upper F right parenthesis or S p ( 2 n , F ) $Sp(2n, F)$ upper S p left parenthesis 2 n comma upper F right parenthesis over a non-Archimedean local field of characteristic zero. We determine all composition factors of a parabolically induced representation of the form ⟨ [ ν a ρ , ν b ρ ] ⟩ ⋊ σ $\langle [ \nu ^{a} \rho , \nu ^{b} \rho ] \rangle \rtimes \sigma $ left angle bracket left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket right angle bracket right normal factor semidirect product sigma , where ⟨ [ ν a ρ , ν b ρ ] ⟩ $\langle [ \nu ^{a} \rho , \nu ^{b} \rho ] \rangle $ left angle bracket left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket right angle bracket denotes the Zelevinsky segment representation of the general linear group attached to the segment [ ν a ρ , ν b ρ ] $[ \nu ^{a} \rho , \nu ^{b} \rho ]$ left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket , and σ $\sigma $ sigma denotes a strongly positive discrete series of G n $G_n$ upper G Subscript n .
Abstract We study the cyclicity of polynomials in Poletsky–Stessin weighted Bergman spaces on various domains in C 2 $\mathbb {C}^2$ double struck upper C squared , including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the parameter range for Poletsky–Stessin weighted Bergman spaces on complete Reinhardt domains, yielding a family of spaces that resemble Dirichlet-type spaces on the unit ball. We highlight the differences in the cyclicity behavior of polynomials in these spaces on the bidisk compared to those studied by Bénéteau et al. Finally, we propose several open problems concerning the structure of cyclic polynomials in these spaces.
Abstract In this article, we investigate several properties of the Volterra-type integral operator V g $V_g$ upper V Subscript g on the Fock space. We give an explicit description of its Berezin range, prove that this range is path connected, and establish several results concerning its convexity. We compute the operator norm, essential norm, Berezin norm, and Berezin number of V g $V_g$ upper V Subscript g . We also derive an exact coefficient representation of its numerical range and obtain a spectral characterization of its numerical radius. In particular, we prove that the operator norm, Berezin norm, Berezin number, and numerical radius coincide if and only if the polynomial expansion of the symbol g contains no linear term. This extremal identity holds even though the operator is non-normal.
We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced C ∗ $C^{*}$ upper C Superscript asterisk -algebra of a product system. We show that for a semigroup dynamical system ( A , P , α ) $(A, P,\alpha )$ left parenthesis upper A comma upper P comma alpha right parenthesis , under reasonable hypotheses (e.g., P is abelian and finitely generated), the reduced crossed product A ⋊ red P $A \rtimes _{\mathrm {red}} P$ upper A right normal factor semidirect product Subscript red Baseline upper P is exact if and only if A is exact. This strengthens our earlier result (Hossain and Sundar, 2026, J. Noncommut. Geom. ), where it was assumed that the action of P on A is by injective endomorphisms. We also compare the groupoid crossed product described in Hossain and Sundar (2026) and the Fell bundle constructed in Rennie et al. (2017, Math. Proc. Cambridge Philos. Soc. , 163, 561–580) for a product system, and show that they are equivalent as Fell bundles.
