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

Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < p,q < infinity and alpha = (alpha(1), alpha(2)) is an element of R-2. In this paper, we introduce multi-parameter Triebel-Lizorkin spaces (F) over dot(p)(alpha,q)(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 [32]. 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 alpha(1) = alpha(2) = 0, q = 2 and 0 < p <= 1, our spaces (F) over dot(p)(alpha,q)(R-m) coincide with the Hardy spaces H-com(p) associated with the composition of two different singular integrals (see [19]). Therefore, our results also give an atomic decomposition of H-com(p). Our work appears to be the first result of atomic decomposition in the Triebel-Lizorkin spaces in the multi-parameter setting.