  Let $(M,J,g,\omega)$ be a K\"ahler 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 K\"ahler 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 K\"ahler manifolds the
kernel of the Dolbeault Laplacian and the kernel of the Bott-Chern Laplacian
coincide.
