Dual Spaces of Multiparameter Local Hardy Spaces

Abstract

In this paper, we study the duality theory of the multiparameter local Hardy spaces $h^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ , and we prove that ${\left(h^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)\right)}^{\ast }=\text{cm}{\text{o}}^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ , where $\text{cm}{\text{o}}^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ are defined by discrete Carleson measure. Moreover, we discuss the relationship among $\text{cm}{\text{o}}^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ , $\text{Li}{\text{p}}^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ , and rectangle $\text{cm}{\text{o}}_{\text{rect}}^p\left({ℝ}^{n_1}\times {ℝ}^{n_2}\right)$ .