Debraj Chakrabarti, Phil Harrington
We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $\cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescibed support and estimates in Sobolev spaces using our method.
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Sobolev regularity of the Bergman and Szegö projections in terms of $\overline{\partial}\oplus\overline{\partial}^{*}$ and $\overline{\partial}_{b}\oplus\overline{\partial}_{b}^{*}$
本文研究了光滑有界伪凸域上的 Bergman 与 Szegö 投影的 Sobolev 正则性，证明了嵌入映射与投影的连续性等价，并指出在 $\mathbb{C}^{2}$ 中 $j_{1}$ 始终正则。
We study the Dirichlet to Neumann operator of the $\overline{\partial}$-Neumann problem, and the relation between the $\overline{\partial}$-Neumann boundary conditions and the Dirichlet to Neumann operator.
Let $Ω$ be a $C^4$-smooth bounded pseudoconvex domain in $\mathbb{C}^2$. We show that if the $\overline{\partial}$-Neumann operator $N_1$ is compact on $L^2_{(0,1)}(Ω)$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial})\cap Dom(\overline{\partial}^*) \to L^2_{(0,1)}(Ω)$ is $L^p$-regular for all $2\leq p<\infty$.
Assume that $Ω_{1}$ and $Ω_{2}$ are two smooth bounded pseudoconvex domains in $\mathbb{C}^{2}$ that intersect (real) transversely, and that $Ω_{1} \cap Ω_{2}$ is a domain (i.e. is connected). If the $\overline{\partial}$-Neumann operators on $Ω_{1}$ and on $Ω_{2}$ are compact, then so is the $\overline{\partial}$-Neumann operator on $Ω_{1} \cap Ω_{2}$. The corresponding result holds for the $\overline{\partial}$-Neumann operators on $(0,n-1)$-forms on domains in $\mathbb{C}^{n}$.
We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev Hodge theory.
Solving the $\partial \overline{\partial}$ for extendable currents without vanishing the boundary cohomology group
本文提出在不消失边界上同调群的情况下，解决可延展电流的$\partial\overline{\partial}$方程问题。
We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $\overline\partial_b$-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.
We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$.
By using a variant Property $(P_q)$ of Catlin, we discuss the relation of small set of weakly pseudoconvex points on the boundary of pseudoconvex domain and compactness of the $\overline{\partial}$-Neumann operator. In particular, we show that if the Hausdorff $(2n-2)$-dimensional measure of the weakly pseudoconvex points on the boundary of a smooth bounded pseudoconvex domain is zero, then the $\overline{\partial}$-Neumann operator $N_{n-1}$ is compact on $(0,n-1)$-level $L^2$-integrable forms.
We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of dbar.
We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-φ})$, under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension. Moreover, we compute the essential spectrum of the complex Laplacian for decoupled weights, $φ(z) = φ_1(z_1) + \dotsb + φ_n(z_n)$, and investigate (non-) compactness of the $\overline\partial$-Neumann operator in this case. More can be said if every $Δφ_j$ defines a nontrivial doubling measure.
Sobolev regularity for the $\overline\partial$-Neumann operator and transverse vector fields
本文通过切向向量场提出新条件，研究$\overline\partial$-Neumann算子在伪凸域上的Sobolev正则性问题。
We develop a method for proving sup-norm and Hölder estimates for $\overline{\partial}$ on wide class of finite type pseudoconvex domains in $\mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $\mathbb{C}^3$, we prove sup-norm and Hölder estimates for all bounded, pseudoconvex domains with real-analytic boundary.
In this paper, making use of the method developed by Catlin and Catlin-Cho,we study the $L^2$-estimate for the mixed boundary conditions on a lunar manifold with the mixed boundary conditions.
$\overline{\partial}$-Estimates on the product of bounded Lipschitz domain
本文在具有Lipschitz边界的有界域乘积上构造积分解算子，求解Cauchy-Riemann方程，并对所有$1<p\le \infty$建立$L^p$-估计。
Let $Ω=\widetildeΩ\setminus \overline{D}$ where $\widetildeΩ$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of $\widetildeΩ$ with connected complement in $\widetildeΩ$. We obtain characterizations of pseudoconvexity of $\widetildeΩ$ and $D$ through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of $\widetildeΩ$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\widetildeΩ$ and $D$ are p…
For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in $\mathbb C^n$ continue to hold on Kähler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known …
In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact $\overline\partial$-Neumann operator $N_q$ ($1\leq q\leq n-1$)? We prove that a smooth bounded pseudoconvex domain $Ω\subseteq\mathbb{C}^3$ with a one-dimensional complex manifold $M$ in its boundary has a noncompact Neumann operator on $(0,1)$-forms, under the additional assumption that $bΩ$ has finite regular D'Angelo $2$-type at a point of $M$, improving previous results of Fu, …
In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new unexpected phenomena. In particular we prove that the Dolbeault isomorphism fails to hold for the Dolbeault cohomology with prescribed support
Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an $L^2$-resolution for the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$. As an application, we obtain a Grauert-Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If $X$ is a Gorenstein space with canonical singularities, then we get also an $L^2$-representation of the flabby cohomology of the structure sheaf $\mathc…
On a compact complex manifold $X$, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if $X$ satisfies the $\partial\overline{\partial}$-Lemma.
On Weighted $L^{p}$-Sobolev Estimates for Solutions of the $\overline{\partial}$-equation on Lineally Convex Domains of Finite Type and Application
本文研究有限型线性凸域上 \\(\overline{\partial}\\)-方程解的加权 \\(L^{p}\\)-Sobolev 估计，并据此建立有限型凸域加权 Bergman 投影在一般距离幂权重下的相应估计。
On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates
本文为多种乘积域上的 $(0,q)$ 形式构造了 $\overline\partial$ 同伦公式，所得同伦算子能同时给出对所有 $k$ 和 $1<p<\infty$ 具有最优 Sobolev 正则性的 $\overline\partial$ 方程解。
On a smooth, bounded pseudoconvex domain $Ω$ in $\mathbb{C}^n$, to verify that Catlin's Property ($P$) holds for $bΩ$, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we consider the support of the Levi core, $S_{\mathfrak{C}(\mathcal{N})}$, a subset of the infinite type points, and show that Property ($P$) holds for $bΩ$ if and only if it holds for $S_{\mathfrak{C}(\mathcal{N})}$. Consequently, if Property ($P$) holds on $S_{\mathfrak{C}(\mathcal{N})}$, then the $\overline{\partial}$-Neumann operator $N_1$ is compact on $Ω$.
Diederich--Fornæss index and global regularity in the $\overline{\partial}$--Neumann problem: domains with comparable Levi eigenvalues
研究拟凸域的Diederich–Fornæss指数与∂̄–Neumann问题全局正则性的关联，证明当Levi形式本征值q₀–和可比较且该指数为1时，相关算子在Sobolev范数下正则，二维域中该指数为1则蕴含问题全局正则。
On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds
本文研究复流形中具有C²边界的a_q域上∂-方程解的正则性，在特定Levi特征值条件下证明了解在闭包上存在并获得1/2阶导数增益。
This article chronicles a development that started around 1990 with \cite{BoasStraube91}, where the authors showed that if a smooth bounded pseudoconvex domain $Ω$ in $\mathbb{C}^{n}$ admits a defining function that is plurisubharmonic at points of the boundary, then the $\overline{\partial}$--Neumann operators on $Ω$ preserve the Sobolev spaces $W^{s}_{(0,q)}(Ω)$, $s\geq 0$. The same authors then proved a further regularity result and made explicit the role of D'Angelo forms for regularity (\cite{BoasStraube93}). A few years later, Kohn (\cite{Kohn99}) initiated a quantitative study of the re…
In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.
An $L^2$-$\partial\overline\partial$-Lemma on a class of complete Kähler manifolds
本文研究了完备凯勒流形上的 $L^2$-$\partial\overline\partial$-引理，通过分析霍奇拉普拉斯算子的谱间隙性，推广了经典凯勒流形上的 $\partial\overline\partial$-引理。
By establishing a unified estimate of the twisted Kohn-Morrey-Hörmander estimate and the $q$-pseudoconvex Ahn-Zampieri estimate, we discuss variants of Property $(P_q)$ of Catlin and Property $(\widetilde{P_q})$ of McNeal on the boundary of a smooth pseudoconvex domain in $\mathbb{C}^n$ for certain high level forms. These variant conditions on the one side, imply $L^2$-compactness of the $\overline{\partial}$-Neumann operator on the associated domain, on the other side, are different from the classical Property $(P_q)$ and Property $(\widetilde{P_q})$. As an application of our result, we show …
We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\overline{\partial}$-Neumann operators $N_q, \overline{\partial}^* N_{q}$, and $\overline{\partial} N_{q}$ are bounded on $L^p$ spaces for some values of $p$ greater than 2.
Many fundamental results of pluripotential theory on the quaternionic space $\mathbb{H}^n$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators $d_0$ and $d_1$ and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of $ (d_0d_1u)^n$. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternioni…
Let $V$ be a compact and irreducible complex space of complex dimension $v$ whose regular part is endowed with a complete Hermitian metric $h$. Let $π:M\rightarrow V$ be a resolution of $V$. Under suitable assumptions on $h$ we prove that $$H^{v,q}_{2,\overline{\partial}}(\operatorname{reg}(V),g)\cong H^{v,q}_{\overline{\partial}}(M),\ q=0,...,v.$$ Then we show that the previous isomorphism applies to the case of Saper-type Kähler metrics and to the case of complete Kähler metrics with finite volume and pinched negative sectional curvatures.
The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=γ_0$ and $\partial_ν\partial_z\partial_{\overline{z}}w=γ$ on $\mathbb{T}$ (the unit circle), $\frac{1}{2πi}\int_{\mathbb{T}}w_{ζ\overlineζ}(ζ)\frac{dζ}ζ=c$, where $\partial_ν$ denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz-Pick type inequalities and Landau type theorem for solutions to the Dirichlet-Neumann problem.
Let $Ω$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $φ$ be a holomorphic function on $Ω$ that is $C^2$-smooth on the closure of $Ω$. We prove that if $H_{\overlineφ}$ is in Schatten $p$-class for $p\leq 2n$ then $φ$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $Ω$ is not Hilbert-Schmidt.
A universal $\overline\partial$ solution operator on nonsmooth strongly pseudoconvex domains
本文在$C^2$强拟凸域和$C^{1,1}$强$\mathbb C$-线性凸域上构造了满足Sobolev与Hölder-Zygmund估计的通用$\overline\partial$解算子，并利用交换子新分解证明了负指标Sobolev空间中解算子的存在性及$\frac12$估计。
In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on singular spaces. As one application, we prove Hartogs' extension theorem for $(n-1)$-complete spaces. Another application is the characterization of rational singularities. It is shown that complex spaces with rational singularities behave quite tame with respect to some $\overline\partial$-equation in the $L^2$-sense. More precisely: a singular point is ratio…
In this paper we discuss compactness estimates for the $\bar \partial $-Neumann problem in the setting of weighted $L^2$-spaces on $\mathbb{C}^n.$ For this purpose we use a version of the Rellich - Lemma for weighted Sobolev spaces.
Let $Ω\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(Ω)$ if and only if the Poincaré-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the $\overline{\partial}$-operator to have closed range in $L^{2}(Ω)$. We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of $Ω$ to be infinite dimensional.
We construct a solution operator for $\overline{\partial}$ equation that gains $\frac{1}{2}$ derivative in the fractional Sobolev space $H^{s,p}$ on bounded strictly pseudoconvex domains in $\mathbb{C}^n$ with $C^2$ boundary, for all $1 < p < \infty$ and $ s > \frac{1}{p}$.
We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.
$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface
本文研究具有半正曲率反正则线丛的紧复曲面上，其反正则除子补集的$\overline{\partial}$上同调群$H^1(M,\mathcal{O}_M)$的结构与性质。
We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the $\overline\partial$-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated $\overline\partial$-Neumann operator.
We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of spaces of forms and currents. At last, we discuss the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property.
Suppose $p\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\mathbb{D}$ satisfying $F$ is absolutely continuous and $\dot{F}\in L^p(0, 2π)$, where $\dot{F}(e^{it})=\frac{\mathrm{d}}{\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\|w_z\|_{L^p}$ and $\|\overline{w_{\bar{z}}}\|_{L^p}$, where $1\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\mathbb{D}$, then we show that $w_z$ and $\overline{w_{\bar{z}}}$ are in the Hardy space $H^p$, where $1\leq p\leq\infty$. The corresponding Hardy norm estimates, …
Reduced-quaternion inframonogenic functions on the ball
该数学研究聚焦三维球上的约化四元数下单调函数，证明对应n次齐次多项式的子空间维数为6n+3，并构造出该类平方可积函数希尔伯特空间的显式可计算正交基。
Suppose $α>-1$ and $1\leq p \leq \infty$. Let $f=P_α[F]$ be an $α$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2π)$, where $\dot{F}(e^{iθ}):=\frac{dF(e^{iθ})}{dθ}$. In this paper, we investigate the membership of $f_z$ and $f_{\overline{z}}$ in the space $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$, the generalized Hardy space. We prove, if $α>0$, then both $f_z$ and $f_{\overline{z}}$ are in $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$. If $α<0$, then $f_z$ and $f_{\overline{z}}\in \mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ if and only …
Let $M$ be an $n$-dimensional $d$-bounded Stein manifold $M$, i.e., a complex $n$-dimensional manifold $M$ admitting a smooth strictly plurisubharmonic exhaustion $ρ$ and endowed with the Kähler metric whose fundamental form is $ω=i\partial\overline{\partial}ρ$, such that $i\overline{\partial}ρ$ has bounded $L^\infty$ norm. We prove a vanishing result for $W^{1,2}$ harmonic forms with respect to the Bott-Chern Laplacian on $M$.
Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})$ and $p \in [1,\infty]$. Recently, Chen et al. (J. Geom. Anal., 2021) extended Zhu's results (J. Geom. Anal., 2020) and proved that (i) if $f$ is a harmonic mapping and $1 \leq p < \infty$, then $f_z$ and $\overline{f_{\overline{z}}} \in B^p(\mathbb{D})$, the Bergman spaces of $\mathbb{D}$. Moreover, (ii) under additional conditions as $f$ being harmonic quasiregular m…
We consider the magnetic Laplacian $Δ_{ν,μ}$ on $\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ Δ_{ν,μ}= 4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial \overline{z_j}} +2iν(E+ \overline{E} +n) +2μ(E- \overline{E} ) -(ν^2+μ^2)|z|^2. $$ We show that $Δ_{ν,μ}$ is connected to the sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_ω\mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. We also discuss invariance properties of $Δ_{ν,μ}$ and give some of their explicit spectral properties.
Sobolev and Hölder estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$
本文通过构造全纯支撑函数，证明了 $\mathbb C^2$ 中有限型伪凸域上 $\overline \partial$ 方程在 Sobolev 和 Hölder-Zygmund 空间中的几乎尖锐估计，扩展了先前结果。
A Solution Operator for the $\overline\partial$ Equation in Sobolev Spaces of Negative Index
本文在负指标Sobolev空间中构造了严格拟凸域上$\overline\partial$方程的解算子，通过同伦公式和反导数算子新技术实现了$\frac12$阶导数增益。
Integrability of Koszul connections on complex vector bundles over domains in ${\mathbf C}^n$
本文研究了复向量丛上Koszul连接的可积性，针对强伪凸或具有至少3个负Levi特征值的边界点，给出了解的存在性和正则性结果。
Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial}$ complex, denoted here $\overlineð_{\mathrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\mathrm{dim}(\mathrm{sing}(X))=0$, is proved also for $\overlineð_{\mathrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overli…
Compact convergence, deformation of the $L^2$-$\overline{\partial}$-complex and canonical $K$-homology classes
本文在紧不可约 Hermite 复空间上，利用奇点解消和泛函分析技术，证明了最大 $L^2$-$\overline{\partial}$ 复形的卷起算子所对应的解析 $K$-同调类在解消映射下保持相等，并构造了显式同伦。
Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L_p(0, 2π)$, where $\dot{F}(e^{it})=\frac{d}{dt} F(e^{it})$ and $p\geq 1$. Recently, the author in \cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\leq p< 2$, then $f_{z}$ and $\overline{f_{\overline{z}}}\in \mathcal{B}^{p}(\mathbb{D}),$ the classical Bergman spaces of $\mathbb{D}$ \cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\leq p\leq \infty$, then $f_{z},$ $\overline{f_{\overline…
We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=μ\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_μ(z, z_0)$ where $K^T_μ(z, z_0)$ is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point $z_0\in {\overline{D}}$. In this connection, we develop the theory of the boundary behavior of the ring $Q-$homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations $\overline{\partial}f=μ\,\par…
The first partial derivatives of generalized harmonic functions
本文研究单位圆盘上 $(α,β)$-调和映射的一阶偏导数在广义 Hardy 空间中的归属，分别在 $α+β>0$ 与 $α+β<0$ 情形给出成员关系及刚性结论。
For a simply connected domain $G$, let $\partial_{a}G$ be the set of accessible points in $\partial G$ and let $\partial_{n} G=\partial G-\partial_{a}G$. A point $a\in\partial G$ is called semi-unreachable if there is a crosscut $J$ of $G$ and domains $U$ and $V$ such that $G-J=U\cup V$ and $a\in(\partial_{n} U\cup\partial_{n} V)-J$. We use $\partial_{sn}G$ to denote the set of semi-unreachable points. In this article we show that a univalent analytic function $ψ$ from the unit disk $D$ onto $G$ extends continuously to $\overline D$ if and only if $\partial_{sn}G=\emptyset$. As a consequence, …
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