Recently, Li and Ponnusamy~\cite{LiPonnusamy2025} established the coefficient conjecture proposed by Wang et al.~\cite{Wang2024} for several prominent geometric subclasses of $\mathcal{S}^0_H(K)$, the class of sense-preserving $K$-quasiconformal univalent harmonic mappings in the unit disk. In this paper, we show that the conjecture continues to hold for a class of $K$-quasiconformal harmonic mappings defined via quasi-subordination. Further, we determine the range of $p>0$ for which such mappings belong to the Hardy space ${\bf h}^p$ and the weighted Bergman space $\mathbf{a}^{\mathbf{p}}_{\boldsymbol{\beta}}$, for $\beta>-1$. Our Hardy space result makes significant progress toward a problem posed by Pavlovi\'{c}, while the Bergman space result sharpens the range obtained by Das and Rasila~\cite{DasRasila}, doubling the previously known bounds. In addition, we obtain refined growth and integral mean estimates for the subclass, improving earlier results and providing further evidence toward an open problem raised by Das et al.~\cite{DasRasila2025}. Parallel results are also discussed for odd $K$-quasiconformal harmonic mappings.
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty....
In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if...
R. Alvarado, Efstathios Konstantinos Chrontsios Garitsis· 0 citations
Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence...
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
We prove a H\"ormander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness f...
Der-Chen Chang, Ji Li, Chao-Jie Wen et al.· 2 citations· ⚡1
Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematic...
Yi-Ru-Ya Zhang· 0 citations
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