$$W^{1,2}$$ Bott-Chern and Dolbeault Decompositions on Kähler Manifolds

Abstract

AbstractLet $$(M,J,g,\omega )$$ ( M , J , g , ω ) be a Kähler manifold. We prove a $$W^{1,2}$$ W 1 , 2 weak Bott-Chern decomposition and a $$W^{1,2}$$ W 1 , 2 weak Dolbeault decomposition of the space of $$W^{1,2}$$ W 1 , 2 differential (p, q)-forms, following the $$L^2$$ L 2 weak Kodaira decomposition on Riemannian manifolds. Moreover, if the Kähler metric is complete and the sectional curvature is bounded, the $$W^{1,2}$$ W 1 , 2 Bott-Chern decomposition is strictly related to the space of $$W^{1,2}$$ W 1 , 2 Bott-Chern harmonic forms, i.e., $$W^{1,2}$$ W 1 , 2 smooth differential forms which are in the kernel of an elliptic differential operator of order 4, called Bott-Chern Laplacian.