Let $q\in(0,\infty]$ and $\varphi$ be a Musielak-Orlicz function with uniformly lower type $p_{\varphi}^-\in(0,\infty)$ and uniformly upper type $p_{\varphi}^+\in(0,\infty)$. In this article, the authors establish various real-variable characterizations of the Musielak-Orlicz-Lorentz Hardy space $H^{\varphi,q}(\mathbb{R}^n)$, respectively, in terms of various maximal functions, finite atoms, and various Littlewood-Paley functions. As applications, the authors obtain the dual space of $H^{\varphi,q}(\mathbb{R}^n)$ and the summability of Fourier transforms from $H^{\varphi,q}(\mathbb{R}^n)$ to the Musielak-Orlicz-Lorentz space $L^{\varphi,q}(\mathbb{R}^n)$ when $q\in(0,\infty)$ or from the Musielak-Orlicz Hardy space $H^{{\varphi}}({\mathbb{R}^n})$ to $L^{\varphi,\infty}(\mathbb{R}^n)$ in the critical case. These results are new when $q\in(0,\infty)$ and also essentially improve the existing corresponding results (if any) in the case $q=\infty$ via removing the original assumption that $\varphi$ is concave. To overcome the essential obstacles caused by both that $\varphi$ may not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of $H^{\varphi,q}(\mathbb{R}^n)$, the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.

In this paper, we establish a weighted maximal $L_2$ estimate of operator-valued Bochner-Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy-Littlewood maximal inequality.

Let Ω be a domain of $(\mathbb{R}^n)$ with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation -Δ_p(·)^Nu = 0 in Ω as well as the corresponding inhomogeneous equation -Δ_p(·)^Nu=f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to -Δ_p(·)^Nu= 0 in Ω, we prove the following:
(i) in dimension $n=2$, for any subdomain $U \Subset \Omega$ and any $\beta \geq 0$, one has $|D u|^\beta D u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and moreover, the map $\left(x_1, x_2\right) \rightarrow|D u|^\beta\left(u_{x_1},-u_{x_2}\right)$ is quasiregular in $U$ in the sense that
$\left|D\left[|D u|^\beta D u\right]\right|^2 \leq-C \operatorname{det} D\left[|D u|^\beta D u\right] \quad$ a.e. in $U$.
(ii) in dimension $n \geq 3$, for any subdomain $U \Subset \Omega$ with $\inf _U p(x)>1$ and $\sup _U p(x)<3+\frac{2}{n-2}$, one has $D^2 u \in L_{\text {loc }}^{2+\delta}(U)$ with a quantitative upper bound, and also with a pointwise upper bound
$\left|D^2 u\right|^2 \leq-C$ $\sum\limits_{1 \le i < j \le n} {} $ $\left[u_{x_i x_j} u_{x_j x_i}-u_{x_i x_i} u_{x_j x_j}\right]$ a.e. in $U$.
Here constants $\delta>0$ and $C \geq 1$ are independent of $u$. These extend the related results obtaind by Adamowicz-Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940-1965 (2010)] when $n=2$ and $\beta=0$.