Let $(X,\omega_X)$ be a compact Kaehler manifold. Denote by $\iota:Y\rightarrow X$ the natural embedding, where $Y$ is a submanifold in $X$. Recall that we have the De Rham-Kodaira Hodge decomposition on smooth $(p,q)$-forms, $$\mathcal I=\bar\partial{\bar\partial}*\mathcal G+{\bar\partial}^*{\bar\partial}\mathcal G+\mathcal H,$$ where $\mathcal I$ is the identity operator, $\mathcal H$ is the harmonic projection, and $\mathcal G$ (resp. ${\bar\partial}^*$) is the Green operator (resp. the adjoint of $\bar\partial$-operator) with respect to the (induced) metric on the corresponding (sub)manifold.
My question is that consider the following special pull-back $\iota^*$, do we have the following commutativity on $(p,q)$-forms:
$$\iota^*\circ{\bar\partial}_X^*={\bar\partial}_Y^*\circ\iota^*\,;\tag{1}\label{1}$$ $$\iota^*\mathcal G_X=\mathcal G_Y\circ\iota^*\,?\tag{2}\label{2}$$
Here are some thoughts of mine: Regarding \eqref{1}, recall that ${\bar\partial}^*=-*\circ{\partial}\circ *$, so \eqref{1} holds if $\iota^*$ and $*$ can commute. However, I don't know how to prove this part... (is this property correct?)
Regarding \eqref{2}, in my opinion, it holds if we have \eqref{1}, here is the argument: Let $\phi\in A^{p,q}(X)$, then $$\phi=\mathcal H(\phi)+\Delta_{\bar\partial}\mathcal G(\phi),$$ thus, $$\iota^*\phi=\iota^*\mathcal H(\phi)+\iota^*\Delta_{\bar\partial}\mathcal G_X(\phi)=\mathcal H(\iota^*\phi)+\Delta_{\bar\partial}\underline{\iota^*\mathcal G_X(\phi)}.$$ The underline thus equals to $\mathcal G_Y\iota^*\phi$ thanks to the uniqueness.
Question 3: for this part, am I right?
Any suggests will be greatly appreciated, thanks a lot.