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

Abstract

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