Jun Cao

The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space ${\mathbb{R}}^n$ are well-established, whereas only partial results are known for the non-smooth domains. In this paper, $\Omega$ is a non-smooth domain of ${\mathbb{R}}^n$ that is bounded and uniform. Suppose p, $q\in [1,\infty )$ and $s\in (n{(\frac{1}{p}-\frac{1}{q})}_{+},1)$ with $n{(\frac{1}{p}-\frac{1}{q})}_{+}:=max\{n(\frac{1}{p}-\frac{1}{q}),0\}$. The authors show that three typical types of fractional Triebel–Lizorkin spaces, on $\Omega$: $F_{p,q}^s(\Omega )$, ${\mathring{F}}_{p,q}^s(\Omega )$ and ${\tilde{F}}_{p,q}^s(\Omega )$, defined via the restriction, completion and supporting conditions, respectively, are identical if $\Omega$ is E-thick and supports some Hardy inequalities. Moreover, the authors show the condition that $\Omega$ is E-thick can be removed when considering only the density property $F_{p,q}^s(\Omega )={\mathring{F}}_{p,q}^s(\Omega )$, and the condition that $\Omega$ supports Hardy inequalities can be characterized by some Triebel–Lizorkin capacities in the special case of $1\le p\le q<\infty$.

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the $L^p$-Calder\'{o}n-Zygmund inequalities extend to a reasonable class of non-compact Riemannian manifolds without the assumption of a positive injectivity radius. In the present paper, we give a positive answer for $1<p2$, we complement the study in G\"{u}neysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calder\'{o}n-Zygmund inequality by adding Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role throughout the proofs.

For any $p\in(0,\,1]$, let $H^{\Phi_p}(\mathbb{R}^n)$ be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function $\Phi_p$, defined by setting, for any $x\in\mathbb{R}^n$ and $t\in[0,\,\infty)$, $$ \Phi_{p}(x,\,t):= \begin{cases} \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}} & \qquad \text{when } n(1/p-1)\notin \mathbb{N} \cup \{0\}; \\ \frac{t}{\log(e+t)+[t(1+|x|)^n]^{1-p}[\log(e+|x|)]^p} & \qquad \text{when } n(1/p-1)\in \mathbb{N}\cup\{0\},\\ \end{cases} $$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $H^p(\mathbb{R}^n)$ and its dual. Moreover, $H^{\Phi_1}(\mathbb{R}^n)$ is the prototype appearing in the real-variable theory of general Musielak-Orlicz Hardy spaces. In this article, the authors find a new structure of the space $H^{\Phi_p}(\mathbb{R}^n)$ by showing that, for any $p\in(0,\,1]$, $H^{\Phi_p}(\mathbb{R}^n)=H^{\phi_0}(\mathbb{R}^n) +H_{W_p}^p(\mathbb{R}^n)$ and, for any $p\in(0,\,1)$, $H^{\Phi_p}(\mathbb{R}^n)=H^{1}(\mathbb{R}^n) +H_{W_p}^p(\mathbb{R}^n)$, where $H^1(\mathbb{R}^n)$ denotes the classical real Hardy space, $H^{\phi_0}(\mathbb{R}^n)$ the Orlicz-Hardy space associated with the Orlicz function $\phi_0(t):=t/\log(e+t)$ for any $t\in [0,\infty)$ and $H_{W_p}^p(\mathbb{R}^n)$ the weighted Hardy space associated with certain weight function $W_p(x)$ that is comparable to $\Phi_p(x,1)$ for any $x\in\mathbb{R}^n$. As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.