Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition

Abstract

Abstract Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < p , q < ∞ ${0<p, q<\infty }$ and α = ( α 1 , α 2 ) ∈ ℝ 2 ${\alpha =(\alpha _1,\alpha _2)\in \mathbb {R}^2}$ . In this paper, we introduce multi-parameter Triebel–Lizorkin spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ associated with different homogeneities arising from the composition of two singular integral operators whose weak (1,1) boundedness was first studied by Phong and Stein [Amer. J. Math. 104 (1982), 141–172]. We then establish its atomic decomposition which is substantially different from that for the classical one-parameter Triebel–Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and sufficient conditions for the boundedness of an operator T on the multi-parameter Triebel–Lizorkin type spaces. In the special case of α 1 = α 2 = 0 ${\alpha _1=\alpha _2=0}$ , q = 2 and 0 < p ≤ 1 ${0<p\le 1}$ , our spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ coincide with the Hardy spaces H com p ${H^p_{\mathrm {com}}}$ associated with the composition of two different singular integrals (see []). Therefore, our results also give an atomic decomposition of H com p ${H^p_{\mathrm {com}}}$ . Our work appears to be the first result of atomic decomposition in the Triebel–Lizorkin spaces in the multi-parameter setting.