This question is motivated by the problem of finding heat kernels to use for the renormalization of quantum field theories on manifolds with boundary.

If $(\mathscr{E}, Q)$ is an elliptic complex on a closed manifold $M$, then if one chooses a metric $(\cdot,\cdot)$ for $\mathscr{E}$, one can define the formal adjoint $Q^*$ to $Q$ and the cohomological degree-zero operator $QQ^*+Q^*Q$ is elliptic. Its kernel is precisely the cohomology of $\mathscr{E}$. Moreover, there is a decomposition

$\mathscr{E}= \ker(Q)\cap\ker(Q^*)\oplus \text{Im}Q\oplus \text{Im}Q^*.$

On a compact manifold with boundary, there is a similar decomposition (see Günter Schwarz: Hodge Decomposition - A Method for Solving Boundary Value Problems) for differential forms: a $k$ form can be written uniquely as the sum of a harmonic $k$-form (one which is both $d$- and $\delta$-closed), the boundary of a form with vanishing tangential component on $\partial M$, and the coboundary of a form with vanishing normal component on the boundary.

**Is there a similar statement which applies for a general elliptic complex on $M$?** I would be happy enough to know this for an elliptic complex which is isomorphic to one of the form $(\mathscr{E’}\otimes \Omega^\bullet_{[0,\epsilon)},Q’\otimes 1+1\otimes d)$ near $\partial M$ (after a choice of collar neighborhood for $\partial M$), and the metric on $\mathscr{E}$ is a product metric under this identification. Here, $(\mathscr{E}’, Q’)$ is an elliptic complex on $\partial M$. In this situation, it still makes sense to define the tangential and normal parts of an element $e\in \mathscr{E}$ near $\partial M$. So, more precisely, my question is: in the situation just described, can one write

$\mathscr{E} = \ker Q\cap \ker Q^*\oplus Q\mathscr{T}^{k-1}\oplus Q^* \mathscr{N}^{k+1},$

where $\mathscr{T}^{k-1}$ is the space of elements of degree $k-1$ in $\mathscr{E}$ with vanishing tangential component on the boundary and $\mathscr{N}^{k+1}$ is the space of elements of degree $k+1$ in $\mathscr{E}$ with vanishing normal component on the boundary?