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Abstract

Let $\Phi$ be a concave function on $(0,\infty)$ of strictly critical lower type index $p_{\Phi}\in(0,1]$ and $\omega\in A^{\mathrm{loc}}_{\infty}(\mathbb{R} ^n)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz–Hardy space $h^{\Phi}_{\omega}(\mathbb{R} ^n)$ via the local grand maximal function. Let $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all $t\in(0,\infty)$. We also introduce the $\mathrm{BMO}$-type space $\mathrm{bmo}_{\rho,\,\omega}(\mathbb{R} ^n)$ and establish the duality between $h^{\Phi}_{\omega}(\mathbb{R} ^n)$ and $\mathrm{bmo}_{\rho,\,\omega}(\mathbb{R} ^n)$. Characterizations of $h^{\Phi}_{\omega}(\mathbb{R} ^n)$, including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h^{\Phi}_{\omega}(\mathbb{R} ^n)$, from which we further deduce that for a given admissible triplet $(\rho,\,q,\,s)_{\omega}$ and a $\beta$-quasi-Banach space $\mathcal{B}_{\beta}$ with $\beta\in(0,1]$, if $T$ is a $\mathcal{B}_{\beta}$-sublinear operator, and maps all $(\rho,\,q,\,s)_{\omega}$-atoms and $(\rho,\,q)_{\omega}$-single-atoms with $q<\infty$ (or all continuous $(\rho,\,q,\,s)_{\omega}$-atoms with $q=\infty$) into uniformly bounded elements of $\mathcal{B}_{\beta}$, then $T$ uniquely extends to a bounded $\mathcal{B}_{\beta}$-sublinear operator from $h^{\Phi}_{\omega}(\mathbb{R} ^n)$ to $\mathcal{B}_{\beta}$. As applications, we show that the local Riesz transforms are bounded on $h^{\Phi}_{\omega}(\mathbb{R} ^n)$, the local fractional integrals are bounded from $h^p_{\omega^p}(\mathbb{R} ^n)$ to $L^q_{\omega^q}(\mathbb{R} ^n)$ when $q>1$ and from $h^p_{\omega^p}(\mathbb{R} ^n)$ to $h^q_{\omega^q}(\mathbb{R} ^n)$ when $q\le 1$, and some pseudo-differential operators are also bounded on both $h^{\Phi}_{\omega}(\mathbb{R} ^n)$. All results for any general $\Phi$ even when $\omega\equiv 1$ are new.