Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem

Jöran Bergh; Ragnar Sigurdsson

Home/Archive/arXiv

arXiv · 1902.01134

Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem

Jöran Bergh·Ragnar Sigurdsson

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that a function, which is defined on a union of lines $\mathbb{C} E$ through the origin in $\mathbb{C}^n$ with direction vectors in $E\subset \mathbb{C}^n$ and is holomorphic of fixed finite order and finite type along each line, extends to an entire holomorphic function on $\mathbb{C}^n$ of the same order and finite type, provided that $E$ has positive homogeneous capacity in the sense of Siciak and all directional derivatives along the lines satisfy a necessary compatibility condition at the origin. We are able to estimate the indicator function of the extension in terms of Siciak's weighted homogeneous extremal function, where the weight is a function of the type of the given function on each given line. As an application we prove a generalization of Helgason's support theorem by showing how the support of a continuous function with rapid decrease at infinity can be located from partial information on the support of its Radon transform.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jöran Bergh, Ragnar Sigurdsson. 2019-02-04. Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem. https://arxiv.org/abs/1902.01134

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions

The celebrated Ohsawa--Takegoshi extension theorem for $L^2$ holomorphic functions on bounded pseudoconvex domains in $\mathbb C^n$ is a fundamental result in the fields of complex analysis and algebraic geometry. In 1995, Ohsawa conjectured that the theorem holds more generally on bounded complete Kähler domains in $\mathbb C^n$. Recently, Chen, Wu and Wang confirmed this conjecture in a special case. In this paper, we extend their result to the case of holomorphic sections of twisted canonical bundles over relatively compact complete Kähler domains in Stein manifolds. As an application, we establish a Hartogs-type extension theorem for plurisubharmonic functions across compact complete pluripolar sets. This result complements a classical theorem of Shiffman and may be regarded as a plurisubharmonic analogue of the Skoda--El Mir extension theorem, thereby filling a gap that appears to have remained open in the literature since at least 1985.

math.CV↗

arXiv2026

The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives

We apply the methods of simultaneous uniformization and composition operators on Besov spaces to the Teichmüller space $T^Z$ of circle diffeomorphisms with Zygmund continuous derivatives. As consequences, we obtain the following: (1) a new proof of the correspondence between quasiconformal self-homeomorphisms of the unit disk with complex dilatations of linear decay order and their quasisymmetric extensions to the unit circle with regularity in the Zygmund continuously differentiable class; (2) a real-analytic equivalence of $T^Z$ with the real Banach space of Zygmund continuous functions on the unit circle.

math.CV↗

arXiv2026

The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

Let $\overline Ω$ be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let $h:Z\to \overline Ω$ be a fibre bundle of Hölder-Zygmund class $Λ^r$, $r>0$, which is holomorphic over $Ω$. Assuming that the fibre is an Oka manifold, we prove that every continuous section $f_0:\overline Ω\to Z$ is homotopic to a section $f_1:\overline Ω\to Z$ of class $Λ^r(\overline Ω)$ which is holomorphic on $Ω$. We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder-Zygmund classes on such domains.

math.CV↗