Weighted Local Orlicz-Hardy Spaces with Applications to Pseudo-differential Operators

Let $\mathrm{\Phi }$ be a concave function on $(0,\mathrm{\infty })$ of strictly lower type $p_{\mathrm{\Phi }}\in (0,1]$ and $\omega\in A^{\mathop$\mathrm{l}\mathrm{o}\mathrm{c}$}_{\infty}($\mathbb{R}$^n)$. We introduce the weighted local Orlicz-Hardy space $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$ via the local grand maximal function. Let $\rho (t)\equiv t^{-1}\mathrm{/}{\mathrm{\Phi }}^{-1}(t^{-1})$ for all $t\in (0,\mathrm{\infty })$. We also introduce the $$-type space ${}_{\rho ,\omega }({\mathbb{R}}^n)$ and establish the duality between $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$ and ${}_{\rho ,\omega }({\mathbb{R}}^n)$. Several real-varaiable characterizations of $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$ are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$. As applications, we show that the local Riesz transforms are bounded on $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$, the local fractional integrals are bounded from {\normalsize$h_{{\omega }^p}^p({\mathbb{R}}^n)$} to {\normalsize$L_{{\omega }^q}^q({\mathbb{R}}^n)$} when $q>1$ and from {\normalsize$h_{{\omega }^p}^p({\mathbb{R}}^n)$} to {\normalsize$h_{{\omega }^q}^q({\mathbb{R}}^n)$} when $q\le 1$, and some pseudo-differential operators are also bounded on both $h_{\omega }^{\mathrm{\Phi }}({\mathbb{R}}^n)$. All results for any general $\mathrm{\Phi }$ even when $\omega \equiv 1$ are new.