For appropriate domains $Ω_{1}, Ω_{2}$ we consider mappings $Φ_{\mathbf A}:Ω_{1}\toΩ_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(Ω_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $Φ_{\mathbf A}$. We then show that each subspace is isomorphic to a weighted Bergman space on $Ω_{2}$. This leads to a formula for the Bergman kernel on $Ω_{1}$ as a sum of weighted Bergman kernels on $Ω_{2}$
We compute explicitly the Bergman kernels of all two dimensional monomial polyhedra, a class of domains including the Hartogs triangle and some of its generalizations. The kernel is computed from the representation of such domains as quotient of the product of a disc and punctured disc.
We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ and the related operator $f \mapsto f - f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_1(r,u-u(0))$ in terms of $\|u\|_{h^1}$, where $M_1$ is the integral mean with $p=1$.
We consider Riemann surfaces $Σ$ with $n$ borders homeomorphic to $\mathbb{S}^1$ and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an $n$-fold direct sum of Bergman spaces of the disk. We show that this period mapping is holomorphic and injective.
In this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudconvex domains in $\mathbb{C}^n$. In particular we prove that a Toeplitz operator built using as kernel a weighted Bergman kernel of weight $β$ and integrating against a measure $μ$ maps continuously (when $β$ is large enough) a weighted Bergman space $A^{p_1}_{α_1}(D)$ into a weighted Bergman space $A^{p_2}_{α_2}(D)$ if and only if $μ$ is a $(λ,γ)$-skew Carleson measure, where $λ=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $γ=\frac{1}λ\left(β+\frac{α_1}{p_1}-\frac{α_2}{p_2}\right)$.…
Polynomials constant on a hyperplane and CR maps of spheres
证明了在具有固定个数非负相异单项式的超平面上多项式常数的一个尖锐度界。这个界限由John P. D'Angelo提出，由D'Angelo、Kos和Riehl在二维中证明，由作者在三维中证明。目前的工作建立在这些结果的基础上，以解决所有维度的猜想。我们还给出了一个完整的描述，所有多项式的尺寸4和更高的尖锐的界限获得。结果证明了球面上的单项CR映射在所有维上的尖锐度界。
We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $φ$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_α$. We investigate the optimal $β$ such that the composition operator $\mathcal{C}_φ$ maps $\mathcal{D}_α$ boundedly into $\mathcal{D}_β$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and …
