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

Abstract

Let (M,J,g,omega ) be a Kähler manifold. We prove a W^{1,2} weak Bott-Chern decomposition and a W^{1,2} weak Dolbeault decomposition of the space of W^{1,2} differential (p, q)-forms, following the 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} Bott-Chern decomposition is strictly related to the space of W^{1,2} Bott-Chern harmonic forms, i.e., W^{1,2} smooth differential forms which are in the kernel of an elliptic differential operator of order 4, called Bott-Chern Laplacian.