Search

In this paper, we prove embedding theorems for the Möbius invariant space \mathcal {Q}_p on the open unit ball of \mathbb {C}^n into logarithmic tent spaces in the Bergman metric. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each difference is compact on the weighted Bergman space. Motivated by that work, in this paper, we discuss which difference is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relationship between sampling measure \mu, Berezin transform \tilde {\mu } and r-averaging transform \widehat {\mu }_r on Bergman spaces. Compared with some results of Luecking [Amer. J. Math. 107 (1985), pp. 85–111], our results provide an equivalent description of sampling measures, which reveals the reason why 1/\hat {\mu }_r\in L^\infty or 1/\tilde {\mu }\in L^\infty does not make sure that \mu is a sampling measure on Bergman spaces. [ABSTRACT FROM AUTHOR]

In 1934, G. H. Hardy and J. E. Littlewood calculated [Proc. London Math. Soc. (2) 36 (1934), pp. 516–531] the optimal Cesàro exponent for Hardy spaces. In this paper we calculate it for mixed norm spaces, hence including the Bergman spaces in particular. The main technical challenge lies in the analysis of the example needed for the critical case. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f\in A_t^p(\mathcal {U}) and \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index \alpha, then f\equiv 0, where \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j=1,\ldots, n-1 and \mathcal {L}_n = \frac {\partial }{\partial z_n}. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a "derivative norm", which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space. [ABSTRACT FROM AUTHOR]

In this paper we derive formulas for the adjoints of a class of composition operators with irrational symbol, in particular, the n-th root functions. We discuss these formulas on both the Hardy space and the Bergman space. [ABSTRACT FROM AUTHOR]

The well-known conjecture due to B. Korenblum about the maximum principle in Bergman space Ap states that for 0 < p < ∞ there exists a constant 0 < c < 1 with the following property. If f and g are holomorphic functions in the unit disk 𝔻 such that |f(z)| ≤ |g(z)| for all c < |z| < 1, then llfllAp ≤ llgllAp . Hayman proved Korenblum’s conjecture for p = 2, and Hinkkanen generalized this result by proving the conjecture for all 1 ≤ p < ∞. The case 0 < p < 1 of the conjecture has so far remained open. In this pAper we resolve this remaining case of the conjecture by proving that Korenblum’s maximum principle in Bergman space Ap does not hold when 0 < p < 1. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a new type of the classical Schwarz lemma at the boundary for holomorphic self-mappings of the unit ball in ℂn, and then give the boundary version of the rigidity theorem. [ABSTRACT FROM AUTHOR]

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak Y (q) condition, the complex Green operator Gq is exactly (globally) regular if and only if the Szegö projections Sq-1, Sq and a third orthogonal projection S q+1 are exactly (globally) regular. The projection S'q+1 is closely related to the Szegö projection Sq+1 and actually coincides with it if the space of harmonic (0, q + 1)-forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the ∂-Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain. We also prove an extension of this result to the case of bounded smooth domains satisfying the weak Z(q) condition on a Stein manifold. [ABSTRACT FROM AUTHOR]