= Solution
The <Hermitian metric> induces pointwise Hermitian inner products on the exterior powers of the complex <cotangent bundle>, with different bidegrees orthogonal, and a positive volume form $dV_g$. Thus the global $L^2$ inner product on smooth <differential forms> of type $(p,q)$ is
$$
\langle\alpha,\beta\rangle_{L^2}=\int_M\langle\alpha(x),\beta(x)\rangle_g\,dV_g.
$$
Let $\bar\partial^*$ be the formal adjoint of the <Dolbeault operator>, defined by integration by parts, and let $\Delta_{\bar\partial}=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$. This <Dolbeault Laplacian> preserves bidegree, is elliptic and nonnegative, and has
$$
\langle\Delta_{\bar\partial}\alpha,\alpha\rangle=\|\bar\partial\alpha\|^2+\|\bar\partial^*\alpha\|^2.
$$
On a compact Hermitian manifold without boundary, the <Dolbeault Hodge decomposition on a compact Hermitian manifold> says that the smooth harmonic space $\mathcal H_{\bar\partial}^{p,q}=\ker\Delta_{\bar\partial}$ is finite-dimensional and
$$
A^{p,q}=\mathcal H_{\bar\partial}^{p,q}\mathbin{\oplus^\perp}\Delta_{\bar\partial}A^{p,q}.
$$
More precisely, there are the orthogonal harmonic projection $H$ and a <Dolbeault Green operator> $G$ taking smooth forms to smooth forms, with $G=0$ on harmonics and $\Delta_{\bar\partial}G=G\Delta_{\bar\partial}=I-H$. Elliptic regularity ensures these are statements about smooth forms, not only $L^2$ completions. The Green operator commutes with $\bar\partial$ and $\bar\partial^*$, since the Laplacian does and its inverse is unique on the orthogonal complement of the kernel.
Expanding $\alpha=H\alpha+\Delta_{\bar\partial}G\alpha$ gives
$$
\boxed{A^{p,q}=\mathcal H_{\bar\partial}^{p,q}\oplus\bar\partial A^{p,q-1}\oplus\bar\partial^*A^{p,q+1}.}
$$
The three summands are orthogonal: harmonic forms are killed by both adjoints, and $\langle\bar\partial u,\bar\partial^*v\rangle=\langle\bar\partial^2u,v\rangle=0$. If $\alpha$ is $\bar\partial$-closed, its coexact component is orthogonal to every closed form; taking its inner product with $\alpha$ shows that component has norm zero. Conversely the first two components are closed. Thus \b[$\boxed{\ker\bar\partial=\mathcal H_{\bar\partial}^{p,q}\oplus\bar\partial A^{p,q-1}}$], with out-of-range form spaces taken to be zero.
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