Luca Baracco, Alexander Tumanov
We prove that the tangential Cauchy-Riemann operator has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold $X$. Our result generalizes the known case when $X$ is a Stein manifold.
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In this note, we investigate the supremum and the infimum of the functional $|a_{n+1}|-|a_{n}|$ for functions, convex and analytic on the unit disk, of the form $f(z)=z+a_2z^2+a_3z^3+\dots.$ We also consider the related problem to maximize the functional $|a_{n+1}-a_{n}|$ for convex functions $f$ with $f"(0)=p$ for a prescribed $p\in[0,2].$
Marx and Strohhäcker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is subordinate to $k_α(z)/z$ if $f$ is convex of order $α$ for $1/2\leα<1$ and conjectured that this is true also for $0<α<1/2.$ Here, $k_α$ is the standard extremal function in the class of normalized convex functions of order $α$ and $k_0(z)=z/(1-z).$ We prove the conjecture and study geometric properties of convex functions of order $α.$ In particular, we prove that …
A note for Carleson measure on bounded $\mathbb{C}$-convex domains
本文研究了 bounded $\mathbb{C}$-凸域上 Carleson 嵌入算子的有界性和紧性，给出了等价刻画及 sharp $\cB$ 条件。
We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore we discuss the relations of such a Bloch type condition with the analogous hyperbolic Lipschitz condition.
In this paper we prove a Radó type result showing that there is no univalent polyharmonic mapping of the unit disk onto the whole complex plane. We also establish a connection between the boundary functions of harmonic and biharmonic mappings. Finally, we show how a close-to-convex biharmonic mapping can be constructed from a convex harmonic mapping.
We solve the $\bar{\partial}$-problem for differential forms in the sens of Whitney.
The d-bar-Neumann operator on (0,q)-forms ($1\le q \le n$) on a bounded convex domain Omega in C^n is compact if and only if the boundary of Omega contains no complex analytic (equivalently: affine) variety of dimension greater than or equal to q.
In this article, we use a Berndtsson-Andersson operator and the Bergman metric in order to solve the $\bar\partial$ equation on convex domains of finite type for forms satisifying a Carleson condition and get norm estimates of the solution in term of the Carleson norm.
In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.
In this paper, we study supnorm and modified Hölder estimates for the integral solution of the di-bar-equation on a class of convex domains of general type in $\C^2$ that includes many infinite type examples.
We construct a metric on a vector bundle $E \rightarrow Z$ restricted to a conic neighbourhood of a relatively compact $1$-convex section of a submersion $Z \rightarrow X$ with at most polynomial poles at the boundary and positive Nakano curvature tensor in bidegree $(p,q).$
In this paper, we consider the Cauchy-Riemann equation $\bar\partial u= f$ in a new class of convex domains in $\C^n.$ We prove that under $L^p$ data, we can choose a solution in the Lipschitz space $Λ_α,$ where $α$ is an optimal positive number given explicitly in terms of $p.$
In this note we extend N. Th. Varopoulos result on zero sets of H p functions of strictly pseudo-convex domains in C n to lineally convex domains of finite type.
A bridge connecting convex analysis and complex analysis and $L^2$-estimate of $d$ and $\bar\partial$
本文提出连接复分析与凸分析的方法，利用$\bar\partial$方程的$L^2$估计推导$d$方程结果并证明凸分析相关曲率正性。
We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.
An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \bar{z}$, where $r$ is a rational function of degree $n \geq 2$, has at most $5 (n - 1)$ zeros. In this note we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $r(z) - \bar{z}$ no more than $5 (n - 1) - 1$ zeros can occur. Moreover, we show that $r(z) - \bar{z}$ is regular, if it has the maximal number of zeros.
The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanlinna class on a class of convex domains of infinite type in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq \infty$, for a particular solution of the tangential Cauchy-Riemann equation on the boundaries of these domains.
This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.
These notes grew out of an expose on M. Gromov's paper "Convex sets and Kähler manifolds'' ("Advances in Differential Geometry and Topology,'' World Scientific, 1990) at the DMV-Seminar on "Combinatorical Convex Geometry and Toric Varieties'' in Blaubeuren in April `93. Gromov's paper deals with a proof of Alexandrov--Fenchel type inequalities and the Brunn--Minkowski inequality for finite dimensional compact convex sets and their variants for compact Kähler manifolds. The emphasis of these notes lies on basic details and the techniques from various mathematical areas involved in Gromov's argu…
We consider a class of non-convex cones $V$ in $\mathbb{R}^n$ which can be presented as (not unique) union of convex cones of some codimension $q$ which we call the index of non-convexity. This class contains non-convex symmetric homogeneous cones studied by the first author and his collaborators. For these cones we consider a construction of dual non-convex cones $V^*$ and corresponding non-convex tubes $T$ and define a cohomological Laplace transform from functions at $V$ to $q$-dimensional cohomology of $T$ using the language of smoothly parameterized uCech cohomology. We give a constructio…
In the paper we give a lower estimate for the essential norm of the $\bar{\partial}$-Neumann operator on convex domains and worm domains of Diederich and Fornaess.
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