A unified approach to nonlinear Stein theorems

We study regularity for the variable exponent quasilinear $\mathfrak{p}\left(x\right)$-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 $u \in \mathbb{E}W^{1, \mathfrak{p}(\cdot)}\left(Ω\right)$ of \begin{align*} \operatorname{div}_{\mathbb{E}} \left( \mathfrak{a}(x) \lvert \nabla_{\mathbb{E}} u \rvert^{\mathfrak{p}\left(x\right)-2} \nabla_{\mathbb{E}} u \right) = f \qquad \text{ in } Ω, \end{align*} where $Ω\subset \mathbb{E}^{n}$, $\mathfrak{a}$ is a uniformly positive bounded scalar function and the exponent function $\mathfrak{p}$ is uniformly bounded away from $1$ and $\infty,$ $\nabla_{\mathbb{E}}u$ is continuous in $Ω$ as soon as $f \in L^{\left( Q_{\mathbb{E}}, 1\right)}\left(Ω\right)$ and $\mathfrak{a}, \mathfrak{p}$ satisfies some conditions regarding the summability of their mean-oscillations. Here $\mathbb{E}^{n}$ is either the Heisenberg group $\mathbb{H}_{n}$ or the Euclidean space $\mathbb{R}^{n}$ and $\operatorname{div}_{\mathbb{E}}$, $\nabla_{\mathbb{E}}$, $Q_{\mathbb{E}}$ 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 $\mathfrak{a}$ and $\mathfrak{p}$ 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.