Triebel-Lizorkin Spaces of Para-Accretive Type and a Tb Theorem

Abstract

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 .

Singular integral operators on Triebel–Lizorkin spaces of para-accretive type

“…which was introduced by Han [9] for p, q > 1, by Deng and Yang [4] for p, q 1, denoted as b −1Ḟ 0 p,q , and by Lin and Wang [15] for larger ranges of p and q. Here is our main result in this article.…”

“…One interest is to consider the boundedness of operators on Hardy spaces, Triebel-Lizorkin spaces or Besov spaces (cf. [5,8,[10][11][12][15][16][17]19,21,22]). The other interests include considering nonconvolution type kernel (e.g.…”

“…Recently, the authors [15] used a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality to characterize homogeneous Triebel-Lizorkin spaces of para-accretive typeḞ α,q b,p . A necessary and sufficient condition of singular integral operators which is bounded fromḞ 0,q 1,p toḞ 0,q b,p , n n+ε < p 1 and n n+ε < q 2 with the regularity exponent ε of the kernel, is also derived in [15].…”

“…A necessary and sufficient condition of singular integral operators which is bounded fromḞ 0,q 1,p toḞ 0,q b,p , n n+ε < p 1 and n n+ε < q 2 with the regularity exponent ε of the kernel, is also derived in [15]. In this article, we study theḞ 0,q 1,p −Ḟ 0,q b,p boundedness of singular integral operators for wider ranges of n n+ε < p < ∞ and n n+ε < q ∞.…”

A Characterization of Besov Spaces of Para-Accretive Type and Its Application

“…For any cube Q and 0 λ > , one denotes by Q λ the cube concentric with Q whose each edge is λ times as long. A generalized Plancherel-Pôlya-type inequality for Triebel-Lizorkin spaces was given in [8]. In this section, one proves the following Plancherel-Pôlya-type inequalities in Besov sense.…”