Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.20301

Characterizations of quasihyperbolic John domains and uniform domains in metric spaces

Shu-Jing Gao·Chang-Yu Guo·Manzi Huang·Xiantao Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a recent work of Zhou and Ponnusamy [Ann. Sc. Norm. Super. Pisa Ci. Sci. 2025], the authors studied the following natural question: find sufficient and necessary conditions for a domain $Ω$ in a metric space $X$ to be quasihyperbolic John. It was proved that Gromov hyperbolic John domains are quasihyperbolic John, quantitatively. As an application, they obtained a characterization of uniform domains in Ahlfors regular spaces. In a recent work, using a deep improved characterization of Gromov hyperbolicity, Guo, Huang and Wang [arXiv 2025] proved the quantitative equivalence bteween inner uniformity and the quasihyperbolic John condition in metric doubling spaces. However, the proof does not yield a similar characterization for uniform domains. In this article, we find a new elementary approach to successfully extend the above characterization to uniform domains: a domain $Ω$ in a doubling length space $X$ is uniform if and only if it is linearly locally connected (LLC) and satisfies the ball separation condition, if and only if it is LLC-1 and quasihyperbolic John, quantitatively. This substantially improved the corresponding results of Zhou and Ponnusamy. Our new approach also allows us to give an alternative proof of the inner uniformity result of Guo-Huang-Wang without using the improved characterization on Gromov hyperbolicity.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shu-Jing Gao, Chang-Yu Guo, Manzi Huang, Xiantao Wang. 2025-12-23. Characterizations of quasihyperbolic John domains and uniform domains in metric spaces. https://arxiv.org/abs/2512.20301

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

KEEP EXPLORING

Related papers

arXiv2026

Explicit Estimates for the Bergman Kernel Form

Let $(L,e^{-ϕ})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $ω=i\partial\overline{\partial}ϕ$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,ω\leqω$ and the shortest nonconstant closed geodesic has length at least $2π$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-ω\leq\mathrm{Ric}\,ω\leqω$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{3}{2m}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.

math.CV↗
arXiv2026

The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

math.CV↗
arXiv2026

Solving non-oscillatory solutions of the Hill equation via the Tumura--Clunie method

We consider the Hill equation $f''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}e^{iz})f=0$ ($†$), where $\mathbf{k}\geq 1$ and $\mathbf{l}\geq 0$ are integers and $b_{-\mathbf{l}}$, $\cdots$, $b_{\mathbf{k}}$ are constants such that $b_{\mathbf{k}}\not=0$. We point out that there is a full correspondence between the class of non-oscillatory solutions such that $λ(f)<\infty$ of equation ($†$) and the class of Liouvillian solutions of equation $x^2u''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}x^{i})u=0$ ($‡$). Then this paper has twofold purposes. First, parallel to Kovacic's algorithms to find the Liouvillian solutions of equation ($‡$), we develop the Tumura--Clunie method to find the non-oscillatory solutions of a higher order version of the Hill equation. Second, for the particular Hill equation $f''-(b_{\mathbf{k}}e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{k}}b_{\mathbf{s}}\not=0$, we use the Tumura--Clunie method to determine the non-oscillatory solution $f$ with an additional zero property.

math.CV↗