A discrete transform and Triebel-Lizorkin spaces on the bidisc

Abstract

We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc F ˙ p α q , f ˙ p α q \dot F_p^{\alpha q},\,\dot f_p^{\alpha q} and establishes the boundedness of transform S ϕ : F ˙ p α q → f ˙ p α q {S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q} and T ψ : f ˙ p α q → F ˙ p α q {T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q} . We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of f ˙ p α q \dot f_p^{\alpha q} for 0 > p ⩽ 1 , p ⩽ q > ∞ 0 > p \leqslant 1,\,p \leqslant q > \infty . The atom supports on an open set, not a rectangle. Duality ( f ˙ 1 α q ) ∗ = f ˙ ∞ − α q ′ , 1 q + 1 q ′ = 1 , q > 1 , α ∈ R {(\dot f_1^{\alpha q})^{\ast }} = \dot f_\infty ^{ - \alpha q’},\,\tfrac {1} {q} + \tfrac {1} {{q’}} = 1,\,q > 1,\,\alpha \in R , is established, too. The case for F ˙ p α q \dot F_p^{\alpha q} is similar.