本文研究独立中心化复高斯系数 Taylor 级数所定义的随机解析函数，给出其具有有界平均振荡的新的、在温和条件下最优的充分条件，并据此得到随机高斯 Hankel 矩阵范数新界及反例。
摘要：
我们考虑由具有独立、中心化复高斯系数的 Taylor 级数给出的随机解析函数。我们给出了此类函数具有有界平均振荡的一个新的充分条件。在一个温和的正则性假设下，该条件是最优的。利用 Holland 和 Walsh 的一个定理，我们作为推论给出了随机高斯 Hankel 矩阵范数的一个新界。最后，我们构造了一些例外的高斯解析函数，这些函数尤其反驳了如下猜想：具有有界平均振荡的随机解析函数总是具有消失平均振荡。
英文摘要：
We consider random analytic functions given by a Taylor series with independent, centered complex Gaussian coefficients. We give a new sufficient condition for such a function to have bounded mean oscillations. Under a mild regularity assumption this condition is optimal. Using a theorem of Holland and Walsh, we give as a corollary a new bound for the norm of a random Gaussian Hankel matrix. Finally, we construct some exceptional Gaussian analytic functions which in particular disprove the conjecture that a random analytic function with bounded mean oscillations always has vanishing mean oscillations.
This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in the memoir preprint: arXiv:2103.09748, [1], submitted for consideration for publication.
In this article, by combining appropriate refined Bohr's inequalities with some techniques concerning bounded analytic functions defined in the unit disk, we generalize and improve several Bohr type inequalities for such functions.
We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relat…
In this paper, we establish five new sharp versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing Schwarz function in place of the initial coefficients in the power series representations of the functions involved and thereby, we generalize several related results of earlier authors.
In this paper, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar{z}G(z)+H(z)$, where $G$ and $H$ are analytic in the unit disk $|z|<1$ with $G(0)=H(0)=0$ and $H'(0)=1$. In particular, two of them are sharp while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.
In this paper, we establish several new versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing $φ=\{φ_n(r)\}^{\infty}_{n=0}$ in place of the $\{r^n\}^{\infty}_{n=0}$ in the power series representations of the functions involved with the Bohr sum and thereby introducing a single parameter, which generalize several related results of earlier authors.
Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. In this article, we study a generalized Bohr's inequality for the class of bounded analytic functions defined on the simply connected domain $$ Ω_γ:=\bigg\{z\in \mathbb{C}: \bigg|z+\fracγ{1-γ}\bigg|<\frac{1}{1-γ}\bigg\}, \,\ \text{for } 0\leq γ<1. $$ Part of its applications, we calculate the Bohr-type radii for some known integral operators.
We extend previous results on boundedness of sets of coherent sheaves on a compact Kähler manifold to the relative and not necessarily smooth case. This enlarged context allows us to prove properness properties of the relative Douady space as well as results related to semistability of sheaves such as the existence of relative Harder-Narasimhan filtrations.
