Atomic decompositions of Triebel-Lizorkin spaces with local weights and applications

Abstract

In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces F _p,q ^s,w(ℝ ⁿ with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ ℝ, p ∈ (0, 1] and q ∈ [p,∞). The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in (0, 1]. Finite atomic decompositions for smooth functions in F _p,q ^s,w(ℝ ⁿ are also obtained, which further implies that a (sub)linear operator that maps smooth atoms of F _p,q ^s,w(ℝ ⁿ uniformly into a bounded set of a (quasi-)Banach space is extended to a bounded operator on the whole F _p,q ^s,w(ℝ ⁿ. As an application, the boundedness of the local Riesz operator on the space F _p,q ^s,w(ℝ ⁿ is obtained.

Keywords

Local weight / Triebel-Lizorkin space / Atom / Lusin-Area function / Riesz transform

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