We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb C^n$, and we give a sufficient capacity condition in order to identify noncyclic vectors.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.

We study the strong continuity of semigroups of composition operators on local Dirichlet spaces.

We give some new characterizations for compactness of weighted composition operators $u{{C}_{\varphi }}$ acting on Bloch-type spaces in terms of the power of the components of $\varphi$, where $\varphi$ is a holomorphic self-map of the polydisk ${{\mathbb{D}}^{n}}$, thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.

Let B denote the unit ball in Cn, and ν the normalized Lebesgue measure on B. For α > −1, define Here cα is a positive constant such that να(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For a twice differentiable, nondecreasing, nonnegative strongly convex function ϕ on the real line R, define the Bergman-Orlicz space Aϕ(να) by

In this paper we prove that a function f ∈ H(B) is in Aϕ(να) if and only if where is the radial derivative of f.

Let ${{B}_{N}}$ be the unit ball in ${{\mathbb{C}}^{N}}$ and let $f$ be a function holomorphic and ${{L}^{2}}$-integrable in ${{B}_{N}}$. Denote by $E\left( {{B}_{N}},\,f \right)$ the set of all slices of the form $\Pi \,=\,L\,\cap \,{{B}_{N}}$, where $L$ is a complex one-dimensional subspace of ${{\mathbb{C}}^{N}}$, for which $f{{|}_{\Pi }}$ is not ${{L}^{2}}$-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices.

Let $B\,=\,{{B}_{n}}$ be the open unit ball of ${{C}^{n}}$ with volume measure $v$, $U\,=\,{{B}_{1}}$ and $B$ be the Bloch space on $U.\,{{\mathcal{A}}^{2,\alpha }}(B)$, $1\,\le \,\alpha \,<\infty $, is defined as the set of holomorphic $f\,:\,B\,\to \,C$ for which

$$\int_{B\,}{{{\left| f(z) \right|}^{2}}}{{\left( \frac{1}{\left| z \right|}\,\log \frac{1}{1\,-\left| z \right|} \right)}^{-\alpha }}\,\frac{dv(z)}{1\,-\,\left| z \right|}\,<\,\infty $$

if $0\,<\,\alpha \,<\infty $ and ${{\mathcal{A}}^{2,1}}(B)\,=\,{{H}^{2}}(B)$, the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic $f\,:\,B\,\to \,U$ for which the composition operator ${{C}_{f}}\,:\,B\,\to \,{{\mathcal{A}}^{2,\alpha }}(B)$ defined by ${{C}_{f}}(g)=g\,\text{o}\,f\text{,}\,g\in B$, is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

The aim of this paper is to study small Hankel operators $h$ on the Hardy space or on weighted Bergman spaces,where $\Omega $ is a finite type domain in ${{\mathbb{C}}^{2}}$ or a strictly pseudoconvex domain in ${{\mathbb{C}}^{n}}$. We give a sufficient condition on the symbol $f$ so that $h$ belongs to the Schatten class ${{S}_{p}}$, $1\,\le \,p\,<\,+\infty $.

A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the $n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized.