Allal Ghanmi
We describe the range of of weighted Cauchy transform and its $k$-Bergman projection when action on weighted true poly-Bargmann spaces constituting an orthogonal Hilbertian decomposition of the Hilbert space of Gaussian functions on the complex plane.
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The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the canonical solution to the non-homogeneous $k$-Cauchy-Fueter equation in a weighted $L^2$-space, by establishing the weighted $L^2$ estimate. The weighted $k$-Bergman space is the space of weighted $L^2$ integrable functions annihilated by the $k$-Cauchy-Fueter operator, as the counterpart of the Fock space of weighted $L^2$-holomorphic functions on $\mathbb{C}^n…
The so-called weighted solid Cauchy transform, from inside the unit disc into the complement of its closure, is considered and their basic properties such as boundedness is studied for appropriate probability measures. The action the disc polynomials is explicitly computed and used to describe the range of its restriction on the weighted true poly-Bergman spaces.
We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the $k$-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô--Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman p…
On the surjectivity of the Cauchy-Riemann and Laplace operators on weighted spaces of smooth functions
本文研究了在特定加权空间上Cauchy-Riemann和拉普拉斯算子的满射性，并通过增长条件确定其满射条件。
Using Hörmander $L^2$ method for Cauchy-Riemann equations from complex analysis, we study a simple differential operator $\bar{\partial}^k+a$ of any order (densely defined and closed) in weighted Hilbert space $L^2(\mathbb{C},e^{-|z|^2})$ and prove the existence of a right inverse that is bounded.
We prove estimates in Hölder spaces for some Cauchy-type integral operators representing holomorphic functions in Cartesian and symmetric products of planar domains. As a consequence, we obtain information on the boundary regularity in Hölder spaces of proper holomorphic maps between symmetric products of planar domains.
A bicomplex proportional fractional $(\vartheta,φ)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function
本文首次定义了基于双曲值函数 Riemann-Liouville 导数的双复数比例分数加权 Cauchy-Riemann 算子，并证明了其分数 Borel-Pompeiu 公式。
Best approximations for the weighted combination of the Cauchy--Szegö kernel and its derivative in the mean
本文研究单位圆盘Hardy空间$H^1$中的极值逼近问题，给出由内函数和复数参数化的Cauchy-Szegö核及其导数加权组合在$H^1_0$子空间下最佳逼近的显式公式，并刻画了对应的极值函数。
In this paper we study the boundedness and compactness characterizations of the commutator of Cauchy type integrals $\mathcal C$ on a bounded strongly pseudoconvex domain $D$ in $C^n$ with boundary $bD$ satisfying the minimum regularity condition $C^{2}$ based on the recent result of Lanzani-Stein and Duong-Lacey-Li-Wick-Wu. We point out that in this setting the Cauchy type integral $\mathcal C$ is the sum of the essential part $\mathcal{C}^\sharp$ which is a Calderón-Zygmund operator and a remainder $\mathcal R$ which is no longer a Calderón-Zygmund operator. We show that the commutator $[b, …
Weighted endpoint bounds for the Bergman and Cauchy-Szegő projections on domains with near minimal smoothness
研究近最小光滑性域上Bergman和Cauchy-Szegő投影的加权端点界，证明了弱型$(1,1)$性质并推导出加权Kolmogorov和Zygmund不等式。
The problem of describing the range of a Bergman space B_2(G) under the Cauchy transform K for a Jordan domain G was solved by Napalkov (Jr) and Yulmukhametov. It turned out that K(B_2(G))=B_2^1(C\bar G) if and only if G is a quasidisk; here B_2^1(C\bar G) is the Dirichlet space of the complement of \bar G. The description of K(B_2(G)) for an integrable Jordan domain is given in [S. Merenkov, "On the Cauchy transform of the Bergman space", Mat. Fiz. Anal. Geom., 7 (2000), no. 1, 119-127]. In the present paper we give a description of K(B_2(G,ω)) analogous to the one given in [S. Merenkov, "On …
In the setting of quaternionic Heisenberg group $\mathscr H^{n-1}$, we characterize the boundedness and compactness of commutator $[b,\mathcal C]$ for the Cauchy--Szegö operator $\mathcal C$ on the weighted Morrey space $L_w^{p,\,κ}(\mathscr H^{n-1})$ with $p\in(1, \infty)$, $κ\in(0, 1)$ and $w\in A_p(\mathscr H^{n-1}).$ More precisely, we prove that $[b,\mathcal C]$ is bounded on $L_w^{p,\,κ}(\mathscr H^{n-1})$ if and only if $b\in {\rm BMO}(\mathscr H^{n-1})$. And $[b,\mathcal C]$ is compact on $L_w^{p,\,κ}(\mathscr H^{n-1})$ if and only if $b\in {\rm VMO}(\mathscr H^{n-1})$.
Finite-sheeted Cauchy operator at rational corners
研究平面楔形区域上开口角为π的有理倍数的柯西奇异积分算子，通过覆盖变换将其分解为区间柯西变换的有限叶组合，并分析其函数空间性质和端点展开。
We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's i…
In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.
The Hardy number and the Bergman number of a planar domain are equal
本文证明任意平面区域的Hardy数与Bergman数相等，从而改进Karafyllia的定理，揭示预设值域函数在Hardy空间与加权Bergman空间中的隶属关系可相互关联。
Weighted Kernel Functions on Planar Domains
本文研究平面区域上加权 Szegő 核与 Garabedian 核随权函数的变化，证明 Ramadanov 型定理，分析零点性质、加权 Ahlfors 映射边界行为及其与经典核函数的关系。
Weighted Szegő Kernels on Planar Domains
该研究探讨平面区域上加权Szegő核与Garabedian核的性质，通过加权Kerzman-Stein公式推导边界光滑性，分析权接近1时的性质传递，并证明约化Bergman核可表为无加权Szegő核的有理组合，拓展了相关核函数体系。
The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity
本文证明树加权平面地图在圆堆积和黎曼均匀化嵌入下收敛于$\sqrt{2}$-LQG，并附带证明自然路径收敛于SLE$_8$。
A linear topological invariant for weighted spaces of holomorphic functions
本文研究由权函数系统定义的带状区域上急速衰减全纯函数Fréchet空间的线性拓扑不变量$(Ω)$，通过权函数系统的显式条件给出完全刻画，并应用于向量值光滑函数加权空间上Cauchy-Riemann算子的满射性研究。
In this article, we consider two classes of weighted Hardy spaces on products of planar domains and their corresponding kernel functions, and we prove product versions of Saitoh's conjecture related to the two classes of weighted Hardy spaces.
The commutator of the Cauchy--Szegő Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted regularity
针对$\boldsymbol{\text{ℂ}^n}$中边界为$C^2$类的有界强拟凸域，研究了加权柯西-塞格投影交换子$[b, S_ω]$在$L^p(bD, Ω_p)$中的紧性与有界性（含显式界），并刻画了$p=2$时交换子$[b,S_{Ω_2}]$在对应空间的相关性质。
We realize the relative discrete series of a weighted $L^2$-space on a bounded symmetric doamin as kernels of invariant Cauchy-Riemann operator, and thus as the spaces of nearly holomorphic functions.
Weighted versions of Saitoh's conjecture in fibration cases
该研究引入平面区域及其乘积纤维化上的广义Hardy空间，研究对应核函数，证明了纤维化情形下Saitoh猜想的若干加权版本。
Weighted estimates for the Bergman projection on planar domains
本文通过 Riemann 映射研究了单连通平面区域上 Bergman 投影的加权估计，提出了获取完整 Békollè-Bonami 型加权估计及弱型 $(1,1)$ 估计所需的正则性条件，并结合共形映射与二进调和分析技术完成证明。
Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance
该文研究对偶 Hardy 空间中的 Cauchy 变换，构造 Möbius 不变下界函数，证明其刚性并关联 Berezin 变换，为椭圆给出渐近尖锐范数下界。
The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity
该研究证明，从临界Boltzman测度采样的树加权平面图在Tutte嵌入下，当边界长度趋于无穷时收敛于参数$\gamma=\sqrt{2}$的Liouville量子引力圆盘，且其上的简单随机游走收敛于平面布朗运动。
The Cauchy--Szegö Projection for domains in $\mathbb C^n$ with minimal smoothness: weighted theory
针对最小光滑性（$C^2$ 边界）的有界强伪凸域，利用定量外推方法，证明了 Cauchy--Szegö 投影在 $A_p$-测度加权 $L^p$ 空间上的显式有界性，并识别了 $p=2$ 时可定义该投影的 Hardy 空间类。
We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2ϕ})$ where $ϕ$ is a subharmonic function with $Δϕ$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $Δϕ$.
Mobius invariant weighted reduced quaternionic Qp-modules
本文引入由约化四元数 Cauchy-Riemann 算子诱导的加权约化四元数模，并证明其加权共形不变性。
We first prove a Cauchy's integral theorem and Cauchy type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection and conformal invariant properties.
The $L^2$-Norm of the Cauchy transform on circular annuli
本文通过傅里叶模分解和加权Hardy算子，计算了圆环上Cauchy变换的L²范数，并给出了特征值和极值解的表达式。
Weighted Bergman Kernels on Planar Domains
本文研究了平面区域上加权Bergman核的边界行为及其与经典核的关系，并探讨了由特定权函数诱导的加权Bergman度量下圆盘到多元柱等区域的全纯等距性质。
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