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

Abstract

Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < , < ∞ and = ( 1 , 2 ) ∈ ℝ 2 . In this paper, we introduce multi-parameter Triebel-Lizorkin spaceṡ , (ℝ ) associated with di erent homogeneities arising from the composition of two singular integral operators whose weak (1, 1) boundedness was rst studied by Phong and Stein [32]. We then establish its atomic decomposition which is substantially di erent from that for the classical one-parameter Triebel-Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and su cient conditions for the boundedness of an operator on the multi-parameter Triebel-Lizorkin type spaces. In the special case of 1 = 2 = 0, = 2 and 0 < ≤ 1, our spaceṡ , (ℝ ) coincide with the Hardy spaces com associated with the composition of two di erent singular integrals (see [19]). Therefore, our results also give an atomic decomposition of com . Our work appears to be the rst result of atomic decomposition in the Triebel-Lizorkin spaces in the multi-parameter setting.

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

“…Furthermore, we have proved the boundedness of multi-parameter pseudo-differential operators and Fourier integral operators on such spaces. Multi-parameter local Hardy space theory can be extended to the setting of multi-parameter local Triebel-Lizorkin and Besov spaces as done in the classical multi-parameter Hardy spaces [11,14,16,42].…”

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

“…Recently, the authors of [9,25] established the boundedness of singular integral operators on multi-parameter Triebel-Lizorkin and Besov-Lipschitz spaces. More recently, the atomic decomposition and dual spaces for multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular operators studied by Phong and Stein [32] were given in [7,6].…”

The boundedness of operators on weighted multi‐parameter mixed Hardy spaces

“…[32], the authors introduced a theory of discrete Calderón reproducing formula and Littlewood–Paley analysis and then developed the implicit multi‐parameter Hardy space theory associated with the flag singular integrals. This discrete Littlewood–Paley theory is particularly useful in dealing with the Hardy spaces with or without weights, no matter in one‐parameter setting or multi‐parameter setting [6, 7, 9–12, 15, 27, 30, 31, 33–35, 37, 38].…”