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

Abstract

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

80 pages, Dissertationes Math. (Rozprawy Mat.) (to appear)