Relations between product and flag Triebel-Lizorkin spaces

Nagel, Ricci and Stein proved that product kernels are finite sums of flag kernels in the Euclidean space. We show that the product Triebel-Lizorkin space is the intersection of two flag Triebel-Lizorkin spaces. This extends a main result in [Chang D-C, Han Y, Wu X. Relations between product and flag Hardy spaces. J Geom Anal. 2021;31(7):6601–6623]. As an application, we provide a new proof of the boundedness of product singular integral operators on product Triebel-Lizorkin spaces.