Search

Suppose D is a bounded strongly pseudoconvex domain in $${{\mathbb {C}}}^n$$ with smooth boundary, and let $$\rho $$ be its defining function. For $$1< p-1$$ , we show that the weighted Bergman projection $$P_\alpha $$ is bounded on $$L^p(D, |\rho |^\alpha dV)$$ . With non-isotropic estimates for $$\overline{\partial }$$ and Stein’s theorem on non-tangential maximal operators, we prove that bounded holomorphic functions are dense in the weighted Bergman space $$A^p(D, |\rho |^\alpha dV)$$ , and hence Hankel operators can be well defined on these spaces. For all $$1

Given some regular weight $$\omega $$ on the unit disk $$\mathbb {D}$$, let $$L^p_\omega $$ be the space of all Lebesgue measurable functions on $$\mathbb {D}$$ for which $$\begin{aligned} \Vert f\Vert _{L^p_\omega }=\left( \int _{\mathbb {D}} |f(z)|^p \omega (z)\mathrm{d}A(z)\right) ^{\frac{1}{p}}

Let $$F^{2}_\varphi $$ be the generalized Fock space defined as $$\begin{aligned} F^2_\varphi =\left\{ f {\text { holomorphic in }} {{\mathbf {C}}^n}: \Vert f\Vert _{2, \varphi }= \left( \int _{{{{\mathbf {C}}^n}}}\left| f(z) e^{-\varphi (z)}\right| ^{2} dv(z)\right) ^\frac{1}{2}

Given \({\varphi\in \verb"C"^2(\textbf{C}^n)}\) satisfying \({dd^{c}\varphi\simeq \omega_0}\), 0 < p < ∞, let \({F^p(\varphi)}\) be the generalized Fock space of all holomorphic functions f on \({{\mathbf C}^n}\) for which the Fock norm $$\|f\|_{p, \varphi}=\left(\,\int_{{\mathbf C}^n} \left|f(z)\right|^{p}e^ {-p\varphi(z)}dv(z)\right)^{\frac{1}{p}} < \infty. $$ While \({\varphi(z)=\frac{1}{2}|z|^2}\), \({F^{p}(\varphi)}\) is the classical Fock space Fp. In this paper, for all possible 0 < p,q < ∞ we characterize those positive Borel measures μ on \({{\mathbf C}^n}\) for which the induced Toeplitz operators Tμ are bounded (or compact) from one generalized Fock spaces \({F^p(\varphi)}\) to another \({F^q(\varphi)}\). With symbols \({g\in BMO}\), we obtain Zorborska’s criterion for boundedness (or compactness) of Toeplitz operators Tg on Fp, our work extends the known results on F2. Toeplitz operators on p-th Fock space with 0 < p < 1 have not been studied before, even in the simplest case that \({\varphi(z)=\frac{1}{2}|z|^2}\). Our analysis shows a significant difference between Bergman spaces and Fock spaces.

Let Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, 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 $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. 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) .

Let Ω ⊂ Rn be a bounded convex domain with C2 boundary. Given 0 < p, q ≤ ∞ and a normal weight function φ (r ), let Hp,q,φ be the harmonic mixed norm space on Ω. In this paper we prove that Gleason's problem (Ω, a , Hp,q,φ ) is always solvable for any reference point a ∈ Ω. Also, Gleason's problem for the harmonic φ -Bloch (little φ -Bloch) space is solvable. The parallel results for the holomorphic functions on bounded convex domains in Cn are obtained. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We characterize those holomorphic symbols ϕ: D → D for which the induced composition operator Cϕ: Bω → Bμ (respectively, Bω,0 → Bμ,0) is bounded or compact, where D is the unit disc in the complex plane C, ω is a normal function on [0, 1) and μ is a non-negative function on [0, 1) with μ(tn) > 0 for some sequence satisfying limn→∞tn= 1.

We define an extended Cesàro operator Tg with holomorphic symbol g in the unit ball B of Cn. For a large class of weights w we characterize those g for which Tg is bounded (or compact) from Bergman space Lpa,w(B) to Lqa,w(B), 0

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn as T g ( f ) ( z ) = ∫ 0 1 f ( t z ) ℜ g ( t z ) d t t , f ∈ H ( B ) , z ∈ B . Where ℜ g ( z ) = ∑ z j ∂ g ∂ z j is the radial derivative of g. In this paper, the author characterizes g for which Tg is bounded (or compact) on the Bloch space B and the little Bloch space B0.

We define an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of C n as T g (f)(z) = ∫ 1 0 f(tz)R g (tz)dt/t, f ∈ H(B), z ∈ B, where R g (z) = Σ n j=1 z j ∂ f /∂ zj is the radial derivative of g. In this paper we characterize those g for which T g is bounded (or compact) on the mixed norm space H p,q (ω).

Let (?) = {x∈Rn; λ(x)0} be a bounded domain with smooth boundary. For e0 small enough, 0p≤∞, 0q∞, s-1, m∈N, and x0∈(?) fixed, it is proved that, for function f harmonic on (?),andwhere Mp(f, r) is the integral mean of f on (?)(?)r, and gradj f is the gradient of f with order j.

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn as T g ( f ) ( z ) = ∫ 0 1 f ( t z ) ℜ g ( t z ) d t t , f ∈ H ( B ) , z ∈ B . Where ℜ g ( z ) = ∑ z j ∂ g ∂ z j is the radial derivative of g. In this paper, the author characterizes g for which Tg is bounded (or compact) on the Bloch space B and the little Bloch space B0.

For a large class of weights ϕ, we characterize the holomorphic symbols g, for which the extended Cesaro operators Tg acting on the weighted Bergman space A 2 ϕ (B) are in the Schatten (or Schatten-Herz) class in the unit ball of C n .

This paper studies the iterated commutators on mixed norm spaces L 2 ( ϕ ) characterizing the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded by using the Bergman geometry, properties of holomorphic functions and related analysis.