A unified approach to nonlinear Stein theorems

Abstract

We study regularity for the variable exponent quasilinear p(x)-Laplace type equation on domains in Heisenberg groups and Euclidean spaces. We establish borderline continuity estimates for the appropriate first order derivatives of solutions. More precisely, we show that for any weak solution u ∈ 𝔼W^{1, p(·)}(Ω) of div_𝔼 ( a(x) lvert ∇_𝔼 u rvert^p(x)−2 ∇_𝔼 u ) = f in Ω, where Ω⊂ 𝔼ⁿ, a is a uniformly positive bounded scalar function and the exponent function p is uniformly bounded away from $1$ and ∞, ∇_𝔼u is continuous in Ω as soon as f ∈ L^{( Q_𝔼, 1)}(Ω) and a, p satisfies some conditions regarding the summability of their mean-oscillations. Here 𝔼ⁿ is either the Heisenberg group ℍ_n or the Euclidean space ℝⁿ and div_𝔼, ∇_𝔼, Q_𝔼 stands for the corresponding divergence, gradient and homogeneous dimension, respectively. We treat the elliptic and subelliptic cases in a unified manner using Euclidean techniques and our conditions on a and p are new and weaker than all the known sufficient conditions even in the Euclidean case. However, all the known sufficient conditions imply our conditions, achieving yet another unification.

Semiparametric Bernstein-von Mises theorems from Stein's method
Paul Rosa
arXiv29 Sep 2026PreprintarXiv licenceFull textStatisticsarXiv
A unified approach for estimation of unnormalized models via Stein operators
Mitsuki Nagai, Keisuke Yano
arXiv21 Oct 2026PreprintCC BY 4.0Full textStatisticsarXiv
A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform
Lingxiao Zhang
arXiv2 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions
Isabella Lewis, Ian Short
arXiv23 Oct 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv
Calabi-Bernstein type theorems for entire solutions of certain nonlinear elliptic equations in the plane
Michael Bildhauer, Martin Fuchs
arXiv9 Oct 2026PreprintCC0Full textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{mallick2026unified,
  title = {{A unified approach to nonlinear Stein theorems}},
  author = {Arka Mallick and Swarnendu Sil},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30949},
  eprint = {2609.30949},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30949},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - A unified approach to nonlinear Stein theorems
AU  - Arka Mallick
AU  - Swarnendu Sil
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30949
UR  - https://arxiv.org/abs/2609.30949
AB  - We study regularity for the variable exponent quasilinear p(x)-Laplace type equation on domains in Heisenberg groups and Euclidean spaces. We establish borderline continuity estimates for the appropriate first order derivatives of solutions. More precisely, we show that for any weak solution u ∈ 𝔼W^1, p(·)(Ω) of div_𝔼 ( a(x) lvert ∇_𝔼 u rvert^p(x)−2 ∇_𝔼 u ) = f in Ω, where Ω⊂ 𝔼^n, a is a uniformly positive bounded scalar function and the exponent function p is uniformly bounded away from $1$ and ∞, ∇_𝔼u is continuous in Ω as soon as f ∈ L^( Q_𝔼, 1)(Ω) and a, p satisfies some conditions regarding the summability of their mean-oscillations. Here 𝔼^n is either the Heisenberg group ℍ_n or the Euclidean space ℝ^n and div_𝔼, ∇_𝔼, Q_𝔼 stands for the corresponding divergence, gradient and homogeneous dimension, respectively. We treat the elliptic and subelliptic cases in a unified manner using Euclidean techniques and our conditions on a and p are new and weaker than all the known sufficient conditions even in the Euclidean case. However, all the known sufficient conditions imply our conditions, achieving yet another unification.
ER  -

Abstract

Related

Cite