Weighted Sub-Bergman Hilbert spaces in the unit ball of ℂn

In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.

REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.

Characterization of weighted analytic Besov spaces in terms of operators of fractional differentiation

Let $$\mathbb{D}$$ stand for the unit disc in the complex plane ℂ. Given 0 < p < ∞, −1 < λ < ∞, the analytic weighted Besov space $$B_p^\lambda \left( \mathbb{D} \right)$$ is defined to consist of analytic in $$\mathbb{D}$$ functions such that $$\int\limits_\mathbb{D} {\left( {1 - \left| z \right|^2 } \right)^{Np - 2} \left| {f^{\left( N \right)} \left( z \right)} \right|^p d\mu _\lambda \left( z \right) < \infty ,}$$ where dμ λ(z) = (λ + 1)(1 − |z|2)λ dμ(z), $$d\mu (z) = \tfrac{1} {\pi }dxdy$$, and N is an arbitrary fixed natural number, satisfying N p > 1 − λ.We provide a characterization of weighted analytic Besov spaces $$B_p^\lambda \left( \mathbb{D} \right)$$, 0 < p < ∞, in terms of certain operators of fractional differentiation R zα,t of order t. These operators are defined in terms of construction known as Hadamard product composition with the function b. The function b is calculated from the condition that R zα,t (uniquely) maps the weighted Bergman kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha }$$ to the similar (weight parameter shifted) kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha - t}$$, t > 0. We also show that $$B_p^\lambda \left( \mathbb{D} \right)$$ can be thought as the image of certain weighted Lebesgue space $$L^p \left( {\mathbb{D},d\nu _\lambda } \right)$$ under the action of the weighted Bergman projection $$P_\mathbb{D}^\alpha$$.

Gaussian curvature of the Bergman metric with weighted Bergman kernel on the unit disc

In the paper Gaussian curvature of Bergman metric on the unit disc and the dependence of this curvature on the weight function has been studied.