Search

The pluripolar hull of a pluripolar set E in $\mathbb{P}^n$ is the intersection of all complete pluripolar sets in $\mathbb{P}^n$ that contain $E$. We prove that the pluripolar hull of each compact pluripolar set in $\mathbb{P}^n$ is $F_\sigma$. The convergence set of a divergent formal power series $f(z_{0}, \dots,z_{n})$ is the set of all "directions" $\xi \in\mathbb{P}^{n}$ along which $f$ is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in $\mathbb{P}^n$ is the convergence set of some divergent series $f$. The convergence sets on $\Gamma:=\{[1:z:\psi(z)]: z\in \mathbb{C}\}\subset\mathbb{C}^2\subset\mathbb{P}^2$, where $\psi$ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on $\Gamma$ is a convergence set in $\mathbb{P}^2$ if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on $\Gamma$ is a convergence set., Comment: 24 pages

A connected compact subset $E$ of $\mathbb{R}^N$ is said to be a strict Whitney set if there exists a real-valued $C^1$ function $f$ on $\mathbb{R}^N$ with $\nabla f|_E\equiv 0$ such that $f$ is constant on no non-empty relatively open subsets of $E$. We prove that each self-similar arc of Hausdorff dimension $s>1$ in $\mathbb{R}^N$ is a strict Whitney set with criticality $s$. We also study a special kind of self-similar arcs, which we call "regular" self-similar arcs. We obtain necessary and sufficient conditions for a regular self-similar arc $\Lambda$ to be a $t$-quasi-arc, and for the Hausdorff measure function on $\Lambda$ to be a strict Whitney function. We prove that if a regular self-similar arc has "minimal corner angle" $\theta_{\min}>0$, then it is a 1-quasi-arc and hence its Hausdorff measure function is a strict Whitney function. We provide an example of a one-parameter family of regular self-similar arcs with various features. For some value of the parameter $\tau$, the Hausdorff measure function of the self-similar arc is a strict Whitney function on the arc, and hence the self-similar arc is an $s$-quasi-arc, where $s$ is the Hausdorff dimension of the arc. For each $t_0\ge 1$, there is a value of $\tau$ such that the corresponding self-similar arc is a $t$-quasi-arc for each $t>t_0$, but it is not a $t_0$-quasi-arc. For each $t_0>1$, there is a value of $\tau$ such that the corresponding self-similar arc is a $t_0$-quasi-arc, but it is a $t$-quasi-arc for no $t\in [1, t_0)$., Comment: 30 pages, 3 figures

Kernels of integral transforms of the form k(xy) are called Fourier kernels. Hardy and Titchmarsh (6) and Watson (15) studied self- reciprocal transforms with Fourier kernels on the positive half-real line R+ and obtained nice characterizing conditions and other properties. Neverthe- less, the Fourier transform has an inherent complex, unitary structure and by treating it on the whole real line more interesting properties can be ex- plored. In this paper, we characterize all unitary integral transforms on L 2 (R) with Fourier kernels. Characterizing conditions are obtained through a natu- ral splitting of the kernel on R+ and R−, and the conversion to convolution integrals. A collection of concrete examples are obtained, which also include "discrete" transforms (that are not integral transforms). Furthermore, we ex- plore algebraic structures for operator-composing integral kernels and identify a certain Abelian group associated with one of such operations. Fractional powers can also be assigned a new interpretation and obtained.

We study the convergence of a formal power series of two variables if its restrictions on curves belonging to a certain family are convergent. Also analyticity of a given C 1 function f is proved when the restriction of f on analytic curves belonging to some family is analytic. Our results generalize two known state- ments: a theorem of P. Lelong and the Bochnak-Siciak Theorem. The questions we study fall into the category of "Osgood-Hartogs- type" problems.

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed., 12 pages

We study the fixed point sets of holomorphic self- maps of a bounded domain in C n . Specifically we investigate the least number of fixed points in general position in the domain that forces any automorphism (or endomorphism) to be the identity. We have discovered that in terms of this number one can give the necessary and sufficient condition for the domain to be biholomor- phic to the unit ball. Other theorems and examples generalize and complete previous results in this area, especially the recent work of Jean-Pierre Vigue.

Let H n be the metric space of all bounded domains in C n with the metric equal to the Hausdorff distance between boundaries of domains. We prove that the dimension of the group of automorphisms of domains is an upper semicontinuous func- tion on H n . We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in H n .

It is well-known that a real analytic manifold X admits a complexification XC , a complex manifold that contains X as the fixed point set of an antiholomorphic involution. This can be seen as follows.The transition functions defining the manifold X are real-analytic local diffeomorphisms of Rn. The Taylor expansions of these transition functions can be considered as local biholomorphisms of Cn, hence they serve as transition functions of a complex manifold. The germ of the complexification XC is unique. Every sufficiently small tubular neighborhood Ω of X in the tangent bundle TX admits a real analytic diffeomorphism into XC that fixes X . Therefore a sufficiently small tubular neighborhood ofX in TX has a complex structure and can be considered as a complexification of X . In general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoXC . There have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofX in TX . With additional datum of a real analytic Riemannian metric g on X a canonical complex structure can be specified for sufficiently small tubular neighborhoods Ω of X in TX (see [GS, LS, S1]). There is a unique complex structure on Ω such that the map f(σ + iτ) = (τγ′(σ))γ(σ), σ + iτ ∈ C, is holomorphic, wherever it is defined

In its proof, a bijective holomorphic mapping σ :Ω → B with σ(q)= 0 has been constructed. However, no explicit arguments showing the complex analyticity of σ−1 had been presented. Indeed, the main purpose of this article is to give it a concise proof. On the other hand, we present it as a separate article since we believe that the arguments here are of a separate interest, in regard to the following unsolved problem

Let $\Gamma$ be a $C^\infty$ curve in $\Bbb{C}$ containing 0; it becomes $\Gamma_\theta$ after rotation by angle $\theta$ about 0. Suppose a $C^\infty$ function $f$ can be extended holomorphically to a neighborhood of each element of the family $\{\Gamma_\theta \}$. We prove that under some conditions on $\Gamma$ the function $f$ is necessarily holomorphic in a neighborhood of the origin. In case $\Gamma$ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\Bbb{C}$.

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.

The disc-hull of a set is the union of the set arid all HiX discs whose boundaries lie in the set. We give an example of a, compact set in C2 whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin. The polynomial hull of a compact set X c C'" is the set X of all points x E C"' at which the inequality IP(x)l : U --* C,7 is a non-constant map whose components are in H' (U), its range d>(U) is called an H?-disc, parametrized by d>. If lim.,/1(4>(rei0)) E X for almost all e'O on the unit circle T, then 4)(U) is an H'-disc whose boundary lies in X." They further define the disc-hull D(X) to be the union of X and all Hc-discs whose boundaries lie in X. Because of the maximum principle, D(X) C X. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X) is always compact for a compact set X C C". Below we answer this question negatively by constructing a counter-exaimple in c2 1. Define w {z E U: Rez > 4 Let Ro: U -> W be the Riemann map satisfying o(+i) = 1 + 23i and ,o(1) = 1. Therefore Re,o(e&0) = 2 for Ree0o ,cn(0) = 0. 2. Let X {((, ?7) E C2 ( E T, q E F(}, where the fiber F, is defined as follows. 1( = T for Re( > 0, 1T( U for ( ?i, and 1( {f01(() 'n E N} U {0} for ReX D? (0) (0, 0); therefore (0, 0) E D(X). Received by the editors August 31, 1999. 2000 Mathematics Subject Classification. Primary 32E20.

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different., Comment: 10 pages

Open Access. Click on the DOI link for free access to the article at the publisher's website.

We show in this paper that every domain in a separable Hilbert space, say $\cH$, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of $\cH$. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [KIK] and subsequent improvement of Byun/Gaussier/Kim [BGK] in the infinite dimensions., 13 pages

It is a result of classical function theory (see [FiF; Les; Mas; PeL; S]) that if f : U → U is a conformal self-mapping of a plane domain that fixes three distinct points then f(ζ) ≡ ζ. The purpose of the present paper is to put this result into a geometrically natural context and to extend it to higher-dimensional domains and manifolds. For an examination of fixed point questions from a slightly different point of view, we refer the reader to the work of Vigue (see e.g. [V1; V2]). The third-named author thanks Robert Burckel for early discussions of this topic and for basic references.

We give examples of non-homogeneous bounded domains in C which can exhaust no other domains. And we prove that every bounded domain in C with C boundary can exhaust the ball.

The (projective) convergence set of a divergent formal power series $f(x_{1},...,x_{n})$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in \mathbb{C}^{n}$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges absolutely near $t=0$. We prove that every countable union of closed complete pluripolar sets in $\PP^{n-1}$ is the convergence set of some divergent series $f$. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional results of A. Sathaye, P. Lelong, N. Levenberg and R.E. Molzon, and of J. Rib\'{o}n are thus generalized., Comment: 17 pages