Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） 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 Fs,w p,q（Rn） are also obtained, which further implies that a （sub）linear operator that maps smooth atoms of Fs,w p,q（Rn） uniformly into a bounded set of a （quasi-）Banach space is extended to a bounded operator on the whole Fs,w p,q（Rn） As an application, the baundedness of the local Riesz operator on the space Fs,w p,q（Rn） is obtained.
