arXiv subjects
Publications and source records attributed to Steven G. Krantz.
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of areas of complex geometric analysis. Relations with other parts of mathematics are described.
The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications.
We provide a simple method for obtain boundary asymptotics of the Poisson kernel on a domain in $\RR^N$.
We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $Ω$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of this methodology, theorems of Calderón type about local boundary behavior on a set of positive measure may be proved in a new and more natural way. We also study the question of nontangential boundedness (on a set of positive measure) versus admissible boundedness. Under suitable hypotheses, these two conditions are shown to be equivalent.
Let $Ω$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families'' arguments in the construction of the limit biholomorphism.
We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.
Many important journal functions would be lost if the mathematical community replaced all paper journals with electronic media. Electronic media are useful for some purposes, but they will not be the basis for a publishing revolution in the near future.
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
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.
We give an explicit description of hyperbolic Reinhardt domains D in C^2 such that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii) D intersects at least one of the coordinate complex lines $\{z_1=0\}$, $\{z_2=0\}$, and (iii) D has noncompact automorphism group. We also give an example that explains why such a setting is natural for the case of hyperbolic domains and an example that indicates that the situation in C^n for n greater than or equal to 3 is essentially more complicated than that in C^2.
We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.
The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary-value problem belonging to a class of problems first introduced by Višik and Eskin, and by Boutet de Monvel.
Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.
The authors study Hardy spaces, of arbitrary order, on a space of homogeneous type. This extends earlier work that treated only $H^p$ for $p$ near 1. Applications are given to the boundedness of certain singular integral operators, especially on domains in complex space.
We give an example of a bounded, pseudoconvex, circular domain in ${\mathbb C}^3$ with smooth, real-analytic boundary and non-compact automorphism group, which is not biholomorphically equivalent to any Reinhardt domain.
The purpose of this bulletin board is to collect problems in higher dimensional complex analysis. We are interested both in basic research questions as well as interactive questions with other fields and sciences. We encourage everybody to submit problems to the list. This includes not only those coming up in your own work, but also others- maybe well known and classical- that you see missing, but that you think workers in the field should be aware of. Not only are we searching for basic research type questions in several complex variables, we also solicit questions exploring relations to other mathematical fields, one complex variable, partial differential equations, differential geometry, dynamics, etc. and to other sciences such as physics, engineering, biology etc. While some questions fall rather naturally into one of the subject areas in this problem list and may lead to a publishable paper, other questions may be non- specific or of a transient or technical quality. Thus we have a section called ``Scratchpad'' for conversational questions, vaguely formulated questions, or questions to which you may hope to get a quick answer. In the ``open prize problems'' section, you are welcome to offer a nice little prize to whoever does it. The miscellaneous section is for announcements of conferences, jokes, remarks on the general state of the field, etc.
We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.