6.2. Weighted Hölder spaces [02HV]

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6.2. Weighted Hölder spaces

We work on the Riemannian $4$ –manifold $(M_{\epsilon},g_{\epsilon})$ constructed in Section 5. The aim of this subsection is to introduce weighted Hölder spaces and prove a weighted Schauder estimate for the operator $D=d^{\ast}+2\,d^{+}$ associated with the metric $g_{\epsilon}$ .

For $R_{0}$ sufficiently large and $\epsilon=\epsilon(R_{0})$ sufficiently small we define a weight function $\rho_{\epsilon}$ as follows: we set

(6.7)

\rho_{\epsilon}=\begin{cases}\epsilon&\mbox{if }\rho_{j}\leq R_{0}\epsilon,\\ \rho_{j}&\mbox{if }2R_{0}\epsilon\leq\rho_{j}\leq\rho_{0},\\ \epsilon&\mbox{if }\rho_{i}\leq R_{0}\epsilon,\\ \rho_{i}&\mbox{if }2R_{0}\epsilon\leq\rho_{i}\leq\rho_{0},\\ 1&\mbox{if }\rho_{j},\rho_{i}\geq 2\rho_{0}\mbox{ for all }j=1,\dots,8,i=1,\dots,n,\end{cases}

and let $\rho_{\epsilon}$ interpolate smoothly and monotonically between the various regions. By abuse of notation we think of $\rho_{\epsilon}$ as defined both on $M_{\epsilon}$ and on $M^{\textup{gh}}_{\epsilon}$ or its double cover $P|_{\mathcal{U}_{\epsilon}}$ .

Definition 6.8.

For each $\delta\in\mathbb{R}$ , $k\in\mathbb{Z}_{\geq 0}$ and $\alpha\in(0,1)$ define the weighted Hölder norm $C^{k,\alpha}_{\delta}$ by

\|a\|_{C^{k,\alpha}_{\delta}}=\sum_{j=1}^{k}{\|\rho_{\epsilon}^{-\delta+j}\nabla^{j}a\|_{C^{0}}}+\text{sup}_{d(x,y)<\text{inj}\,g_{\epsilon}}{\min{\left\{\rho_{\epsilon}(x)^{-\delta+k+\alpha},\rho_{\epsilon}(y)^{-\delta+k+\alpha}\right\}}\frac{|\nabla^{k}a(x)-\nabla^{k}a(y)|}{|x-y|^{\alpha}}}.

Here all norms and covariant derivatives are computed with respect to the metric $g_{\epsilon}$ and $\nabla^{k}a(x)$ and $\nabla^{k}a(y)$ are compared using parallel transport along the unique geodesic connecting $x$ and $y$ . Similarly set $\|a\|_{C^{0}_{\delta}}=\|\rho^{-\delta}a\|_{C^{0}}$ .

The following simple estimate for products in $C^{0,\alpha}_{\delta-1}$ will be used to control the non-linearities.

Lemma 6.9.

For every $\delta<1$ there exists a constant $C>0$ independent of $\epsilon$ such that

\|u\,v\|_{C^{0,\alpha}_{\delta-1}}\leq C\epsilon^{\delta-1}\|u\|_{C^{0,\alpha}_{\delta-1}}\|v\|_{C^{0,\alpha}_{\delta-1}}.

Proof.

From the definition of the $C^{0,\alpha}_{\delta-1}$ –norm it is immediate to check that

\|u\,v\|_{C^{0,\alpha}_{\delta-1}}\leq C\|\rho_{\epsilon}^{\delta-1}\|_{C^{0}}\|u\|_{C^{0,\alpha}_{\delta-1}}\|v\|_{C^{0,\alpha}_{\delta-1}}.

Since $\rho_{\epsilon}\geq c\,\epsilon$ and $\delta-1<0$ the result follows. ∎

We now consider the operator $D=d^{\ast}+2\,d^{+}$ acting on $1$ –forms of class $C^{1,\alpha}_{\delta}$ . We prove the following weighted Schauder estimate.

Proposition 6.10.

For every $\delta\in\mathbb{R}$ there exists a constant $C>0$ independent of $\epsilon$ such that

\|a\|_{C^{1,\alpha}_{\delta}}\leq C\left(\|Da\|_{C^{0,\alpha}_{\delta-1}}+\|a\|_{C^{0}_{\delta}}\right).

Proof.

In order to prove this estimate it is convenient to cut the manifold $M_{\epsilon}$ in various pieces and analyse the geometry separately in each of them. The global estimate follows by combining the “local” estimates obtained in each of these pieces.

Consider first the region $\rho_{j}\leq 2R_{0}\epsilon$ for some $j=1,\dots,8$ . The rescaled metric $\epsilon^{-2}g_{\epsilon}$ is isometric to a compact region in the $D_{m_{j}}$ ALF space $(M_{j},g_{M_{j}})$ .

Now, given a $1$ –form $a$ on $M_{\epsilon}$ , restrict $a$ to the region $\rho_{j}\leq 2R_{0}\epsilon$ and define $\tilde{a}=\epsilon^{-1-\delta}a$ . The standard Schauder estimate for the elliptic operator $D$ associated with the metric $g_{M_{j}}$ is

\|\tilde{a}\|_{C^{1,\alpha}}\leq C\left(\|D\tilde{a}\|_{C^{0,\alpha}}+\|\tilde{a}\|_{C^{0}}\right).

Since $|\tilde{a}|_{\epsilon^{-2}g_{\epsilon}}=\epsilon^{-\delta}|a|_{g_{\epsilon}}$ and the norms $\nabla\tilde{a}$ and $D\tilde{a}$ are related in a similar way to those of $\nabla a$ and $Da$ , the weighted Schauder estimate follows immediately.

The same argument can be applied in the region $\rho_{i}\leq 2R_{0}\epsilon$ : the role of $M_{j}$ is now played by a small perturbation (cf. the beginning of Section 5.2) of the $A_{k_{i}-1}$ ALF space $N_{i}$ .

Consider now the transition region $R_{0}\,\epsilon\leq\rho_{j}\leq\rho_{0}$ for some $j=1,\dots,8$ . We can work on the double cover $H^{2m_{j}-4}$ and restrict to $\mathbb{Z}_{2}$ –invariant forms. Scaling by $\epsilon$ as above we reduce to consider the restriction of $H^{2m_{j}-4}$ to the region $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ in $\mathbb{R}^{3}$ endowed with a metric

g=\left(1+\epsilon\lambda_{j}+\frac{m_{j}-2}{\rho}\right)g_{\mathbb{R}^{3}}+\left(1+\epsilon\lambda_{j}+\frac{m_{j}-2}{\rho}\right)^{-1}\theta^{2}+O(\rho^{-3})+O(\epsilon^{3}\rho^{2}).

Moreover, after rescaling the weight function $\rho_{\epsilon}$ coincides with the radial function $\rho$ on $\mathbb{R}^{3}$ .

Fix a number $\sigma\in(0,1)$ . For each point $x$ let $\pi(x)$ be its image in $\mathbb{R}^{3}$ and set $R=\sigma\rho(x)$ . Up to changing $R_{0}$ and $\rho_{0}$ into $(1-\sigma)R_{0}$ and $(1+\sigma)\rho_{0}$ we can assume that $B_{R}(\pi(x))$ is contained in the annulus $R_{0}\leq\rho\leq\frac{\rho_{0}}{\epsilon}$ and that the restriction of $H^{2m_{j}-4}$ to this ball is trivial. We can then work on a “square” $B_{R}\times[-R,R]$ in the universal cover of $H^{2m_{j}-4}|_{B_{R}}$ . Rescaling the metric by $R^{-2}$ , applying standard Schauder estimates, rescaling back and multiplying by $R^{-\delta}$ we obtain

\|a\|_{C^{1,\alpha}_{\delta}(B_{R})}\leq C\left(\|Da\|_{C^{0,\alpha}_{\delta-1}(B_{R})}+\|a\|_{C^{0}_{\delta}(B_{R})}\right).

The case of the region $R_{0}\epsilon\leq\rho_{i}\leq\rho_{0}$ is completely analogous.

Finally, in the region where $\rho_{j},\rho_{i}\geq\tfrac{1}{2}\rho_{0}$ the weight function $\rho_{\epsilon}$ is uniformly equivalent to the constant $1$ and therefore weighted spaces coincide with standard Hölder spaces. Moreover the harmonic function $h_{\epsilon}$ is $C^{\infty}$ –close to the constant $1$ . The metric $g_{\epsilon}$ is therefore $C^{\infty}$ –close to the metric $g_{\infty}=g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ . The Schauder estimate for forms supported in this region is immediate since we can restrict to small balls in the torus $\mathbb{T}$ on which the circle bundle $P$ is trivial and then work on the universal cover, which has bounded geometry. ∎