Note on Duality of Weighted Multi-Parameter Triebel-Lizorkin Spaces

Abstract

We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces $\dot{F}_{p}^{\alpha, q}\left(\omega ; \mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right)$. This space has been introduced and the result

$$\left(\dot{F}_{p}^{\alpha, q}\left(\omega ; \mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right)\right)^{*}=\mathrm{CMO}_{p}^{-\alpha, q^{\prime}}\left(\omega ; \mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right)$$

for 0 < p ⩽ 1 has been proved in Ding, Zhu (2017). In this paper, for 1 < p < ∞, 0 < q < ∞ we establish its dual space $\dot{H}_{p}^{\alpha, q}\left(\omega ; \mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right)$.