Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators

Abstract:

In this paper, we study the duality theory of the multi-parameter Triebel-Lizorkin spaces $\dot F^{\alpha ,q}_{p}(\mathbb R^{m})$ associated with the composition of two singular integral operators on $\mathbb {R}^m$ of different homogeneities. Such composition of two singular operators was considered by Phong and Stein in 1982. For $1<p<\infty$, we establish the dual spaces of such spaces as $(\dot F^{\alpha ,q}_{p}(\mathbb R^{m}))^*=\dot F^{-\alpha ,q’}_{p’}(\mathbb R^{m})$, and for $0<p\leq 1$ we prove $(\dot F^{\alpha ,q}_{p}(\mathbb R^{m}))^*=CMO^{-\alpha ,q’}_{p}(\mathbb {R}^m)$. We then prove the boundedness of the composition of two Calderón-Zygmund singular integral operators with different homogeneities on the spaces $CMO^{-\alpha ,q’}_{p}$. Surprisingly, such dual spaces are substantially different from those for the classical one-parameter Triebel-Lizorkin spaces $\dot {\mathcal {F}}^{\alpha ,q}_p(\mathbb R^{m})$. Our work requires more complicated analysis associated with the underlying geometry generated by the multi-parameter structures of the composition of two singular integral operators with different homogeneities. Therefore, it is more difficult to deal with than the duality result of the Triebel-Lizorkin spaces in the one-paramter settings. We note that for $0<p\leq 1$, $q=2$ and $\alpha =0$, $\dot F^{\alpha ,q}_{p}(\mathbb R^{m})$ is the Hardy space associated with the composition of two singular operators considered in Rev. Mat. Iberoam. 29 (2013), 1127–1157. Our work appears to be the first effort on duality for Triebel-Lizorkin spaces in the multi-parameter setting.