The Einstein equation on a Lorentzian manifold can be considered as an evolutional problem. The initial data is subject to the Einstein constraint equation, which is an underdetermined elliptic system. It is a nontrivial question how the initial data can be localized -- the positive mass theorem implies that it cannot be localized to a compact set. Corvino--Schoen and Carlotto--Schoen developed gluing techniques in compact regions and in conic regions, respectively, to construct initial datas. Following the work of Oh--Tataru, we construct solution operators for the linearized Einstein constraint equations with nice support properties. As applications, we give a new proof of Carlotto--Schoen type gluing with optimal decay rate (and optimal regularity) conjectured by Carlotto and recently resolved by Aretakis--Czimek--Rodnianski using characteristic gluing techniques. We also construct more localized initial data to a degenerate sector, making progress on another open problem by Carlotto. Another application of our method is a new proof of the obstruction-free gluing to exterior Kerr initial data recently obtained by Czimek--Rodnianski, with optimal range of charges, optimal regularity and weaker decay assumption. This talk is based on joint work with Yuchen Mao and Sung-Jin Oh.