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

In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces F (p,q) (s,w) (a"e (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 a a"e, p a (0, 1] and q a [p,a). 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) (a"e (n) are also obtained, which further implies that a (sub)linear operator that maps smooth atoms of F (p,q) (s,w) (a"e (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) (a"e (n) . As an application, the boundedness of the local Riesz operator on the space F (p,q) (s,w) (a"e (n) is obtained.