W1 , 2 Bott-Chern and Dolbeault Decompositions on Kähler Manifolds

Let (M, J, g, ω) be a Kähler manifold. We prove a W1 , 2 weak Bott-Chern decomposition and a W1 , 2 weak Dolbeault decomposition of the space of W1 , 2 differential (p, q)-forms, following the L2 weak Kodaira decomposition on Riemannian manifolds. Moreover, if the Kähler metric is complete and the sectional curvature is bounded, the W1 , 2 Bott-Chern decomposition is strictly related to the space of W1 , 2 Bott-Chern harmonic forms, i.e., W1 , 2 smooth differential forms which are in the kernel of an elliptic differential operator of order 4, called Bott-Chern Laplacian.