We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

We investigate integral means over spherical shell of holomorphic functions in the unit ball $\mathbb {B}_n$ of $\mathbb {C}^n$ with respect to the weighted volume measures and their relation with the weighted Hadamard product. The main result of this paper has many consequences which improve some well-known estimates related to the Hadamard product in Hardy spaces and weighted Bergman spaces.

We obtain sharp ranges of $L^p$ -boundedness for domains in a wide class of Reinhardt domains representable as sublevel sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$ -boundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is $L^p$ -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases .

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$

${\it\mu}$

$|{\it\mu}|(\mathbb{D})<\infty$

$B_{p}$

$0<p\leq 1$

${\rm\Lambda}_{t}$

$t>1$

It has been shown that a holomorphic function $f$ in the unit ball ${{\mathbb{B}}_{n}}$ of ${{\mathbb{C}}_{n}}$ belongs to the weighted Bergman space $A_{\alpha }^{p},\,p\,>\,n\,+\,1\,+\alpha $, if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\beta }}\,\times \,d{{v}_{\beta }})$, where $\beta \,=\,(p\,+\,\alpha \,-\,n\,-\,1)/2$ and $d{{v}_{\beta }}(z)\,=\,{{(1\,-\,{{\left| z \right|}^{2}})}^{\beta }}\,dv(z)$. In this paper we consider the range $0\,<\,p\,<\,n\,+\,1\,+\,\alpha $ and show that in this case, $f\,\in \,A_{\alpha }^{p}\,(\text{i})$ (i) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| 1\,-\,\left\langle z,\,w \right\rangle \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$, (ii) if and only if the function $\left| f(z)\,-\,f(w) \right|/\left| z\,-\,w \right|$ is in ${{L}^{p}}({{\mathbb{B}}_{n}}\,\times \,{{\mathbb{B}}_{n}},\,d{{v}_{\alpha }}\,\times \,d{{v}_{\alpha }})$. We think the revealed difference in the weights for the double integrals between the cases $0\,<\,p\,<\,n\,+\,1\,+\,\alpha $ and $p\,>\,n\,+\,1\,+\,\alpha $ is particularly interesting.

Let Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which , where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by . In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

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.

We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for the the interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov and others.

It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

For 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.

2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.

We solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.

AMS 2000 Mathematics subject classification: Primary 47B35; 47B47

For a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.