In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive typeḞ α,q b,p , which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a necessary and sufficient condition for theḞ 0,q 1,p −Ḟ 0,q b,p boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M b T M b ∈ WBP is bounded fromḞ 0,q 1,p toḞ 0,q b,p if and only if T b ∈Ḟ 0,q b,∞and T * b = 0 for n n+ε < p ≤ 1 and n n+ε < q ≤ 2, where ε is the regularity exponent of the kernel of T .
We introduce the generalized Carleson measure spaces CMOrα,qthat extend BMO. Using Frazier and Jawerth'sφ-transform and sequence spaces, we show that, forα∈Rand0<p≤1, the duals of homogeneous Triebel-Lizorkin spacesḞpα,qfor1<q<∞and0<q≤1are CMO(q'/p)-(q'/q)-α,q'and CMOr-α+(n/p)-n,∞(for anyr∈R), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel-Lizorkin spaces.
Keywords:Calderón reproducing formula Para-accretive function Paraproduct operator T b theorem Triebel-Lizorkin spaceIn this article, we study the boundedness of singular integral operators acting on TriebelLizorkin spaces of para-accretive type and show a T b theorem on these spaces, which extends previous results in David, Journé and Semmes (1985) [3], Han (1994) [9], Han, Lee and Lin (2004) [10], Han and Sawyer (1990) [12], Lin and Wang (2009) [15], Wang (1999) [22].
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