$W^{1,2}$ 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, 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. We also generalize to the non compact case the well known property that on compact Kähler manifolds the kernel of the Dolbeault Laplacian and the kernel of the Bott-Chern Laplacian coincide.