Carleson duality and endpoint structure of Euclidean flag Triebel--Lizorkin spaces

The main purpose of this paper is to develop the duality theory of the homogeneous Triebel--Lizorkin spaces associated with the Euclidean flag, using the discrete Littlewood--Paley--Stein analysis. After establishing bounded analysis and synthesis maps, we describe the continuous duals by Carleson conditions. In the Hardy spatial range, the representing distributions satisfy an open-set Carleson condition when the sequence exponent is greater than one and a coefficient supremum condition when the sequence exponent is at most one; in the Banach spatial range the dual is a conjugate-exponent or a supremum space. At the supremum sequence endpoint, the closure of the flag test functions is characterized by a pointwise scale tail measured in $L^p$, and a coefficient pairing decomposes every functional on the full space into a regular part and a part that annihilates the closure. By means of a fixed-envelope modulation construction we show that this closure is proper for every finite spatial exponent and that the annihilator is nonzero when the spatial exponent is at least one. The same analysis and synthesis framework also yields complex interpolation, real interpolation in the spatial exponent, the lifting property and the standard mixed-norm embeddings.