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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 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 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 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]

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 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]

Based on recent advancements of reproducing kernels in several complex variables, the theory of rational approximation for weighted Bergman spaces on tube domains over cones is developed. By verifying the boundary vanishing property, the existence of optimal parameters and the corresponding orthogonal functions are obtained. [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]

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and which commutes with a quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a "hyperbolic" composition operator. [ABSTRACT FROM AUTHOR]

The Friedrichs operator of a domain (in \mathbb {C}^n) is closely related to its Bergman projection and encodes crucial information (geometric, quadrature, potential theoretic etc.) about the domain. We show that the Friedrichs operator of a domain has rank one if the domain can be covered by a circular domain via a proper holomorphic map of finite multiplicity whose Jacobian is a homogeneous polynomial. As an application, we show that the Friedrichs operator is of rank one on the tetrablock, pentablock, and the symmetrized polydisc – domains of significance in the study of \mu-synthesis in control theory. [ABSTRACT FROM AUTHOR]

In this paper we prove an isoperimetric inequality for holomorphic functions in the unit polydisc Un. As a corollary we derive an inclusion relation between weighted Bergman and Hardy spaces of holomorphic functions in the polydisc which generalizes the classical Hardy-Littlewood relation Hp ... A2p. Also, we extend some results due to Burbea. [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]

Let \omega and \nu be radial weights on the unit disc of the complex plane such that \omega admits the doubling property \sup _{0\le r<1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}<\infty. Consider the one weight inequality \begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space. [ABSTRACT FROM AUTHOR]

We show that Bergman completeness is not a quasi-conformal invariant for general Riemann surfaces. [ABSTRACT FROM AUTHOR]

Let $M$ be an invariant subspace of the Bergman space $L_a^2(mathbb {D})$ and $S_M$ be the compression of the coordinate multiplication operator $M_z$ to the co-invariant subspace $L_a^2(mathbb {D})ominus M$. The present paper determines when the identity representation of $C^*(S_M)$ is a boundary representation for the Banach subalgebra $mathcal {B}(S_M)$. The paper also considers boundary representations on the co-invariant subspaces of $L_a^2(mathbb {B}_n)$. [ABSTRACT FROM AUTHOR]

The boundedness of the maximal operator on the upper half-plane \Pi ^{+} is established. Here \Pi ^+ is equipped with a positive Borel measure d\omega (y)dx satisfying the doubling property \omega ((0,2t))\leq C\omega ((0,t)). This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A_{\omega }^{p}(\Pi ^{+}). [ABSTRACT FROM AUTHOR]

For the rational power-generalized Hartogs triangles in \mathbb {C}^2, we give a complete characterization of the weak-type regularity of the Bergman projection at the upper and lower endpoints of L^p boundedness. Our result extends work of Huo-Wick for the classical Hartogs triangle by showing that the Bergman projection satisfies a weak-type estimate only at the upper endpoint of L^p boundedness. [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]

We show that the norm in the Hardy space H^p satisfies \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f')(1-r)^{p\left (1-\frac 1q\right)}\,dr+|f(0)|^p\tag {\dagger } \end{equation} for all univalent functions provided that either q\ge 2 or \frac {2p}{2+p}

We prove that a Toeplitz operator with a complex Borel measure symbol whose total variation is Carleson is strongly localized, thus improving the recent localization results from Sadeghi and Zorboska [J. Math. Anal. Appl. 485 (2020), p. 16]. We also show that as a corollary, we get yet another view of some of the previously known results on Toeplitz operators with BMO symbols. [ABSTRACT FROM AUTHOR]

We prove two sharp estimates for the subspace of a standard weighted Bergman space that consists of functions vanishing at a given point (with prescribed multiplicity). [ABSTRACT FROM AUTHOR]

We provide a new characterization (valid for all 0 < p < ∞) of Schatten class membership of Toeplitz operators induced by positive measures acting on weighted Bergman spaces on the unit ball. The result is new even for the case of the unit disk. [ABSTRACT FROM AUTHOR]

In this paper, we identify a large collection of open subsets of the complex plane for which the core of corresponding unbounded Bergman operators is densely defined. This result gives the necessary background to investigate the concept of invariant subspaces, index, and cyclicity in the unbounded case. [ABSTRACT FROM AUTHOR]

In this paper, we first discuss some basic results concerning Toeplitz operators with quasihomogeneous symbols (i.e., symbols being of the form $ e^{iptheta }varphi $, where $varphi $ is a radial function) on the harmonic Bergman space. Then we determine when the product of two Toeplitz operators with quasihomogeneous symbols is a Toeplitz operator. [ABSTRACT FROM AUTHOR]

In this paper, we discuss reducing subspaces of multiplication operators $M_phi $ on the Dirichlet space $mathcal {D}$ defined by a Blaschke product $phi $ with two zeros $a$, $b$ in the unit disk $mathbb {D}$ and show that when $a+b=0$, $M_phi $ has two proper ones; otherwise it has none. This is different from the cases of the Hardy space and the Bergman space. par [ABSTRACT FROM AUTHOR]

We study the relationship between polynomial approximations in the Bergman space of certain simply connected domains in the complex plane and composition operators on the Dirichlet space of the unit disk. In particular, we characterize when a composition operator on the Dirichlet space has dense range, which settles a problem posed by Joseph Cima in 1976. [ABSTRACT FROM AUTHOR]

In both the Bergman space $A^2$ and the Hardy space $H^2$, the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function $g$ in $H^{\infty}$ has the wandering property in $X$, where $X$ denotes either $A^2$ or $H^2$, provided that every $g$-invariant subspace $M$ of $X$ is generated by the orthocomplement of $gM$ within $M$. It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of $H^{\infty}$ also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property. [ABSTRACT FROM AUTHOR]

Let φ be an analytic self-map of the open unit disk D. We study the complex symmetry of composition operators Cφ on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D that is not an elliptic automorphism, we establish that if Cφ is complex symmetric, then either φ (0) = 0 or φ is linear. In the case of weighted Bergman spaces A2α, we find the non-automorphic linear fractional symbols φ such that Cφ is complex symmetric. [ABSTRACT FROM AUTHOR]

We study the Korenblum Maximum Principle on the weighted Fock spaces and the weighted Bergman spaces with exponential weights. First, we give explicit expressions for the upper bounds of Korenblum constants for the weighted Fock spaces. Then, we obtain upper bounds of such constants for the weighted Bergman spaces. Finally, we show a failure of the Korenblum Maximum Principle for weighted Bergman spaces Apα (D), where 0 < p < 1, α > 0. [ABSTRACT FROM AUTHOR]

We obtain Lp regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain Ω with some symmetry properties and generate successor domains in higher dimensions. We prove: If the Bergman kernel on Ω satisfies appropriate estimates, then the Bergman projection on the successor is Lp bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on Lp for 1 < p < ∞. The successor domains need not have smooth boundary nor be strictly pseudoconvex. [ABSTRACT FROM AUTHOR]

We consider non-selfadjoint operator algebras L(G, λ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs G. These algebras may be viewed as non-commutative generalizations of weighted Bergman space algebras or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for L(G, λ) in the case of the single vertex graph with two edges and a suitable choice of left weight function λ. [ABSTRACT FROM AUTHOR]

We characterize those radial doubling weights ω for which the Schwarz-Pick lemma applied to the derivative of any inner function in the norm of the Bergman space Aωp does not cause any essential loss of information. The approach we employ is based on operator theory and leads to a characterization of when the linear average operator ... is bounded from Aωp to Lωp. The characterizing integral condition is selfimproving and therefore T : Aωp → Lωp is bounded if and only if T : Aωp-ε → Lωp-ε is bounded for all sufficiently small ε > 0. This study also reveals the fact that, under appropriate hypothesis on ω, the average operator T : Aωp → Lωp is bounded if and only if the Bergman projection P : Lωp → Lωp is bounded if and only if the classical Hilbert operator H : Lωp+1 ([0, 1)) → Aωp+1 is bounded, where .... [ABSTRACT FROM AUTHOR]

In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol. [ABSTRACT FROM AUTHOR]

We characterize the positive Borel measures such that the differentiation operator of order n ∈ N U {0} is compact from the Hardy space Hp into Lq (μ), 0 < p,q < ∞. [ABSTRACT FROM AUTHOR]

In this note we show that the weighted L²-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space L² (Ω, dμ0) where Ω is a smoothly bounded pseudoconvex domain of finite type in ℂn and μ0 = (-ρ0)r dλ, with λ the Lebesgue measure, r ∈ ℚ+ and ρ0 a special defining function of Ω, are still valid for the Bergman projection of L² (Ω, dμ) where μ = (-ρ)r dλ, with ρ any defining function of Ω and r ∈ R+. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations (for r ∈ ℚ+) are obtained for weighted Lp-Sobolev and Lipschitz estimates in the case of the pseudoconvex domain of finite type in ℂ² (or, more generally, when the rank of the Levi form is ≥ n - 2). [ABSTRACT FROM AUTHOR]

We give a variant of the Bohenblust-Hille inequality which, for certain families of polynomials, leads to constants with polynomial growth in the degree. [ABSTRACT FROM AUTHOR]

For a multiplier pair (X, Y ) we study the closed densely defined operators T on X that commute with all of the multiplications by right multipliers in X. We apply our general results to special cases involving Hp, completions of L∞ [0, 1] with respect to certain norms, and the completion of a II1 factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such T is a "left multiplication". However, we give an example of a closed densely defined operator on the Bergman space that commutes with all multiplications by H∞-functions but is not a multiplication operator. [ABSTRACT FROM AUTHOR]

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szegö and the Bergman kernel on the symmetrized polydisc. [ABSTRACT FROM AUTHOR]

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in Cd. [ABSTRACT FROM AUTHOR]