The Greene-Krantz Conjecture in Dimension Two
We give a proof of the Greene-Krantz conjecture on convex domains in $\CC^2$. Curiously, the proof technique depends on subelliptic estimates for the $\bar{\partial}$ problem.
math.CV↗
arXiv subjects
Steven G. Krantz
Publications and source records attributed to Steven G. Krantz.
On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.
math.CV↗
On the Bergman Projection and the Lu Qi-Keng Conjecture
On a reasonable class of domains in $\CC^n$, we characterize those holomorphic functions which continue analytically past the boundary. Then we give some applications of this result to holomorphic mappings. In addition, some new results about the Lu Qi-Keng conjecture are treated.
math.CV↗
Homage à A. M. Gleason et I. J. Schark
We study the maximal ideal space of $H^\infty(B)$, where $B$ is the unit ball of $\CC^n$. Following the lead of Gleason and Schark, we analyze Gleason parts, fibers, the S\uılov boundary, and other aspects of this Banach algebra. Our work here makes good use of the inner functions construction of Aleksandrov and Hakim/Lø w/Sibony, particularly as formulated by Rudin.
math.CV↗
Restricted Admissible Limit for Domains of Finite Type
We investigate the boundary behavior of holomorphic functions with respect to a family of curves in a domain of finite type. This work is a generalization of uCirka's classical result on the unit ball and it supplements the result by Cima and Krantz on the Lindelöf principle for general domains. See [KRA2] for some recent developments in this subject
math.CV↗
Semicontinuity of the Automorphism Groups of Domains with Rough Boundary
Based on some ideas of Greene and Krantz, we study the semicontinuity of automorphism groups of domains in one and several complex variables. We show that semicontinuity fails for domains in $\CC^n$, $n > 1$, with Lipschitz boundary, but it holds for domains in $\CC^1$ with Lipschitz boundary. Using the same ideas, we develop some other concepts related to mappings of Lipschitz domains. These include Bergman curvature, stability properties for the Bergman kernel, and also some ideas about equivariant embeddings.
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A Direct Connection Between the Bergman and Szegő Kernels
We use Stokes's theorem to establish an explicit and concrete connection between the Bergman and Szegő projections on the disc, the ball, and on strongly pseudoconvex domains.
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On the maximal ideal space of $H^\infty(B)$
We study the corona problem on the unit ball and the unit polydisc in $\CC^n$. We provide affirmative solutions to both problems.
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Canonical kernels versus constructible kernels
We study both canonical reproducing kernels and constructive reproducing kernels for holomorphic functions in $\CC^1$ and $\CC^n$. We compare and contrast the two, and also develop important relations between the two types of kernels. We prove a new result about the relationship between these two kernels on certain domains of finite type.
math.CV↗
Semi-continuity of Automorphism Groups of Strongly Pseudoconvex Domains: the low differentiability case
We study the semicontinuity of automorphism groups for perturbations of domains in complex space or in complex manifolds. We provide a new approach to the study of such results for domains having minimal boundary smoothness. The emphasis in this study is on the low differentiability assumption and the new methodology developed accordingly.
math.CV↗
The Carathéodory and Kobayashi/Royden Metrics by Way of Dual Extremal Problems
We study the Carathéodory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly incisive results are obtained for convex domains.
math.CV↗
Some New Thoughts on Maximal Functions and Poisson Integrals
We study Wiener-type covering lemmas, Hardy-Littlewood-type maximal functions, and convergence theorems on metric spacs. Later we specialize down to a result for the Poisson integral. We show that, in a suitably general setting, these three phenomena are essentially logically equivalent. Along the way we discuss some useful estimates for the Poisson kernel.
math.AP↗
Geometric Properties of Boundary Orbit Accumulation Points
We study the automorphism group action on a bounded domain in $\CC^n$. In particular, we consider boundary orbit accumulation points, and what geometric properties they must have. These properties are formulated in the language of Levi geometry.
math.CV↗
The Automorphism Group of a Domain with an Exponentially Flat Boundary Point
We study the automorphisms group action on a bounded domain in $\CC^n$ having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.
math.CV↗
Topological/Geometric Properties of the Orbit Accumulation Set
We study the set of boundary orbit accumulation points of the automorphism group action on a bounded domain in $\CC^n$. Topological and geometric properties of this set are derived.
math.CV↗
On Limits of Sequences of Holomorphic Functions
We study functions which are the pointwise limit of a sequence of holomorphic functions. In one complex variable this is a classical topic, though we offer some new points of view and new results. Some novel results for solutions of elliptic equations will be treated. In several complex variables the question seems to be new, and we explore some new avenues.
math.CV↗
Deformations of Strongly Pseudoconvex Domains
We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert's ideas about Kobayashi extremal discs, and also has intrinsic interest.
math.CV↗
The Schwarz lemma at the boundary
The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function on the disc. Pick's version of the Schwarz lemma allows one to move the origin to other points of the disc. In the present paper we explore versions of the Schwarz lemma at a boundary point of a domain (not just the disc). Estimates on derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present some new theorems as well.
math.CV↗