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 systematically on modern techniques such as those developed in \cite{FZ2022}. We then give two globalization schemes on bounded, connected domains. For bounded John domains, instead of Reshetnyak's original construction \cite{R19762}, which glues M\"obius transformations on adjacent Whitney cubes, we use Whitney chains and Bojarski's enlarged-cube estimate to obtain a quantitatively sharp Euclidean stability result. Finally, we extend this idea to general bounded, connected open subsets of $\mathbb{R}^n$ via the Lorentzian representation of the M\"obius group, yielding a compactified, weighted stability theorem whose underlying measure is absolutely continuous with respect to Lebesgue measure and has density bounded above by one.
For connected proper metric spaces $X$, equipped with a locally doubling measure supporting a local $p$-Poincar\'e inequality, we completely characterize which compact sets $K$ with positive capacity are removable for bounded $p$-harmonic functions, $p>1$. Similar results are proved also for bounded quasiharmonic funct...
We study $-\Delta_p u = |u|^{q-1}u$ in $\mathbb{R}^N$ with $N>p\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^*-1<q<q_c(N,p)$ only...
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...
Jasbir Parashar, Saminathan Ponnusamy, A. Kaliraj· 0 citations
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homo...
In this article, we study three properties of a bounded $\mathbb{C}$-convex domain $\Omega \subset \mathbb{C}^n$, $n \geq 2$, with smooth boundary, namely Gromov hyperbolicity of its Kobayashi distance, finiteness of the D'Angelo type of $\partial \Omega$, and the existence of a subelliptic estimate for the $\bar{\part...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
Fan-Ze Kong, Xiao-Yu Zeng· 0 citations
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