The estimate will be proved by the standard integration by parts. Since the eigenfunctions $\varphi_{k}$ satisfy

and $\|\varphi_{k}\|_{L^{2}(Y^{3})}=1$, we have that

where $Q_{2K_{0}}$ depends only on the asymptotic bound of $\nabla^{2K_{0}}\xi$. The proof is done.

∎

We will prove that there exists some constant $\eta_{0}<\eta<\eta_{0}+\underline{\delta}/10$ such that

and

where the positive constant $C_{0}>0$ is independent of the index $k$.

We prove (4.136) and (4.137) in two different cases.

In the first case, $k\in\mathbb{Z}_{+}$ satisfies $j_{k}=0$. The fundamental solutions have an explicit form

and

Immediately,

and hence for $\eta>\eta_{0}$,

Similarly,

In the latter case $j_{k}\in\mathbb{Z}_{+}$ and $k\in\mathbb{Z}_{+}$, we will prove the uniform estimates. A crucial point is to apply the monotonicity in Lemma 4.9. In fact,

where $\eta^{\prime}>\eta_{0}$. We choose $\epsilon\in(\underline{\delta}/100,\underline{\delta}/10)$ and denote $\eta\equiv\eta^{\prime}+\epsilon$, then by Lemma 4.9

The proof of (4.136) is done.

Next, for the estimate (4.137),

This completes the proof of the proposition.

∎

The proof follows from the standard elliptic regularity. Indeed, the eigenfunction $\varphi_{k}$ satisfies the elliptic equation

It follows from the standard elliptic regularity that there exists some constant $C>0$ depending only the metric $h_{0}$ such that

Applying the Sobolev embedding theorem,

where $C>0$ depends only on the metric $h_{0}$. The proof is complete. ∎

The proof of the proposition is constructive. The basic strategy is to apply separation of variables to construct a solution to the equation (4.151). Given a function $v$ and for any fixed $z\geq 1$, there is an expansion over the fiber $Y^{3}$,

Separation of variables enables us to construct a formal solution

to the equation (4.151), where $u_{k}$ are the particular solutions in Lemma 4.13. Since a priori the above series is defined in the $L^{2}$-topology along each fiber $Y^{3}\times\{z\}$, we need to verify the higher order convergence of the series, which will indicate that $u$ is a regular solution to (4.151).

First, we will show that the above series converges in the $C^{0}$-topology and thus $u$ is a $C^{0}$-function. The main point is to reduce the uniform convergence to the convergence of certain numerical series involving only in the eigenvalues $\{\Lambda_{k}\}_{k=1}^{\infty}$ of a definite fiber $(Y^{3},h_{0})$. Indeed, Lemma 4.12 guarantees that the solutions satisfy all the conditions in Lemma 4.13. Since we have obtained in Lemma 4.13 the uniform estimate for the ODE solutions $u_{k}$ and also in Lemma 4.14 the uniform estimate for the eigenfunctions, the $L^{2}$-expansion has the following bound,

Since the spectrum of Laplacian $\{\Lambda_{k}\}_{k=1}^{\infty}$ obeys Weyl’s law on $(Y^{3},h_{0})$, it follows that for sufficiently large $k$,

where $C_{0}>0$ depends only on $h_{0}$. Plugging the above asymptotics into (4.155), we have that

and hence

Therefore, $u\in C^{0}(\mathcal{C})$ and $u$ exponentially decays.

Next, we will apply the standard elliptic regularity on the Calabi manifold to show that $u\in C^{2}$ and thus $u$ is a regular solution. For the expansions

we denote by

the partial sums of $u$ and $v$ respectively. Immediately,

For every $\bm{x}\equiv(z,\bm{y})\in\mathcal{C}$, we will apply the elliptic regularity on the ball $B_{2}(\bm{x})\subset\mathcal{C}$ to obtain the higher regularity of $u$. For this purpose, first we prove the following claim.

The proof of the higher order convergence is exactly the same. In fact, we just need to replace $\|v\|_{C^{2K_{0}}}$ with the higher order norm $\|v\|_{C^{2K_{0}+m}}$ with $m\leq K_{0}$. Since $\Delta_{g_{\mathcal{C}}}U_{N}=V_{N}$, the standard $W^{2,p}$- implies that regularity for every $1<p<\infty$, $\|U_{N}\|_{W^{2,p}(B_{1}(\bm{x}))}\leq C_{p,\bm{x}}$. By assumption $v\in C^{3K_{0}}(\mathcal{C})$ with $K_{0}\geq 3$, we have $\|V_{N}\|_{C^{2}(B_{2}(\bm{x}))}\leq C_{\bm{x}}$. Hence the regularity of $u$ will be improved as follows, for every $1<p<\infty$,

Now taking $p>4$ and applying the Sobolev embedding,

which implies that $U_{N}$ converges to a smooth solution $u$. Then applying the standard Schauder estimate and bootstrapping, the statement of the proposition just follows.

∎