3.5. Harmonic analysis I: periodic Euclidean region [0432]

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3.5. Harmonic analysis I: periodic Euclidean region

This Section obtains refined mapping properties of the Euclidean Green operator $\Delta_{a}^{-1}$ on $\mathbb{R}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ , which will be used to correct volume form error away from $\mathfrak{D}$ . The method is similar to Lemma 2.18, and the new technical difficulties are the exponential decay estimate and the growth of the error at large distance. We shall identify $T^{2}$ -invariant functions with functions on the base.

As a preliminary observation, the periodic Newtonian potential on $\mathbb{R}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ equipped with the Euclidean metric $g_{a}$ is given by

G_{a}(\mu_{1},\mu_{2},\eta)=\sum_{n\in\mathbb{Z}}\frac{-1}{4\pi^{2}|(\mu_{1},\mu_{2},\eta+n)|_{a}^{2}}.

Its zeroth Fourier mode is

\bar{G}_{a}(\mu_{1},\mu_{2},y)=\int_{0}^{1}G_{a}dx=-\frac{1}{4\pi A^{1/2}\varrho}.

Up to a factor $A^{1/2}$ this agrees with the Newtonian potential for $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$ . Our real emphasis will be on the second derivatives $\nabla^{2}_{g_{a}}G_{a}$ . Since higher order estimates follow from easy bootstrap arguments, we will focus on absolute estimates.

Lemma 3.17.

For $\varrho\gtrsim A^{1/2}$ ,

|\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}|_{g_{a}}\leq CA^{1/2}\varrho^{-5}.

Proof.

By the mean value inequality

|\int_{\eta-\frac{1}{2}}^{\eta+\frac{1}{2}}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-2}ds-|(\mu_{1},\mu_{2},\eta)|_{a}^{-2}|\leq CA|(\mu_{1},\mu_{2},\eta)|_{a}^{-4},

changing $\eta$ to $\eta+n$ and summing over $n\in\mathbb{Z}$ , we obtain for $\varrho\gtrsim A^{1/2}$ that

\begin{split}&|G_{a}(\mu_{1},\mu_{2},\eta)-\bar{G}_{a}(\mu_{1},\mu_{2},y)|\leq\sum_{n}CA|(\mu_{1},\mu_{2},\eta+n)|_{a}^{-4}\\ \leq&CA\int_{-\infty}^{\infty}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-4}ds\\ \leq&CA^{1/2}\varrho^{-3}.\end{split}

The claim follows from $\Delta_{a}$ -harmonicity and bootstrap arguments. ∎

We can improve this to an exponential decay estimate:

Lemma 3.18.

For $\varrho\gtrsim A^{1/2}$ ,

|\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}|_{g_{a}}\leq CA^{-2}e^{-2\pi A^{-1/2}\varrho}.

Proof.

The basic idea is Fourier analysis in the $x$ -variable combined with $\Delta_{a}$ -harmonicity. The argument is a simpler version of Proposition 3.5, using the barrier method. ∎

Lemma 3.19.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ . Then $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is estimated on $\mathcal{B}^{+}_{\nu}$ by

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\nu\begin{cases}A^{\delta/4}\ell^{\delta}\quad&\ell\lesssim A^{1/2},\\ A^{3\delta/4}\quad&\ell\gtrsim A^{1/2}.\end{cases}

Remark 3.6.

The support cutoff condition $\varrho<A^{1/2}e^{\nu}$ is needed because sources located at exponentially large distance drives up the elliptic constants; this suggests the metric ansatz destabilizes at exponentially large distance (cf. Section 3.10).

Remark 3.7.

The Green operator will propagate the effects out of $\text{supp}(f)$ into the tail region $\{\varrho\geq A^{1/2}e^{\nu}\}$ and the neighbourhood $\{\ell\leq 2A^{-1/4}\}$ of $\mathfrak{D}$ .

Proof.

The basic idea is similar to Proposition 2.18. We analyse the contribution of the source located at $q$ to the convolution integral $\nabla^{2}_{g_{a}}G_{a}*f(p)$ , depending on the spatial separation between $p$ and $q$ . We write $|q|_{a}^{\prime}=\varrho(q),|p|_{a}^{\prime}=\varrho(p)$ .

Suppose $p$ and $q$ do not belong to the same dyadic scale, namely $|p|_{a}^{\prime}\geq A^{1/2}+2|q|_{a}^{\prime}$ or $|q|_{a}^{\prime}\geq A^{1/2}+2|p|_{a}^{\prime}$ . From Lemma 3.17 we easily deduce

|\nabla^{2}_{g_{a}}G_{a}|_{g_{a}}\leq A^{-1/2}\min(|q|_{a}^{\prime-3},|p|_{a}^{\prime-3}),

so the contribution from all such dyadic scales on $\text{supp}(f)$ is bounded by

\begin{split}&CA^{3\delta/4-1/2}(\int_{2|p|_{a}<\varrho<A^{1/2}e^{\nu}}\varrho^{-3}d\text{Vol}_{a}+\int_{A^{1/2}\lesssim\varrho<|p|_{a}^{\prime}/2}|p|_{a}^{\prime-3}d\text{Vol}_{a})\leq CA^{3\delta/4}\nu,\end{split}

where we use $\delta>-3$ to control the source $f=O(A^{\delta/4}\ell^{\delta})$ in $L^{1}$ .

We are left with one dyadic scale $|q|_{a}^{\prime}\sim|p|_{a}^{\prime}\lesssim A^{1/2}e^{\nu}$ . By a similar argument, the contribution from sources at $A^{1/2}\lesssim|p-q|_{a}\lesssim 2|p|_{a}^{\prime}$ is bounded by $CA^{3\delta/4}\nu.$ If $\ell(p)>\frac{1}{2}A^{1/2}$ , then the contribution from sources at $|p-q|_{a}\leq\frac{1}{4}A^{1/2}$ is controlled by $CA^{3\delta/4}$ using standard Schauder theory.

If $\ell(p)\leq\frac{1}{2}A^{1/2}$ and $|p-q|_{a}^{\prime}\lesssim A^{1/2}$ , then for the purpose of estimating the convolution integral we can simply replace the Green kernel $G_{a}$ by $\frac{-1}{4\pi^{2}|(\mu_{1},\mu_{2},\eta)|_{a}^{2}}$ , and correspondingly for their second derivatives. The point is that at this length scale the periodicity effect is secondary, and we are essentially in the same situation as Lemma 2.18 with $-3<\delta<0$ and $\tau=0$ . A careful examination of that argument there, restoring the $A$ -dependence, shows that the contribution of sources inside this region towards $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is bounded by $C(A^{1/4}\ell)^{\delta}.$

Combining the above shows the claim. ∎

Lemma 3.20.

(Exponential decay of higher Fourier modes) In the situation of Lemma 3.19, the higher Fourier modes of $\nabla_{g_{a}}^{2}\Delta_{a}^{-1}f$ admit estimate in the region $\{\ell>A^{1/2}\}\subset\mathcal{B}^{+}_{\nu}$ ,

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}(f-\bar{f})|_{g_{a}}\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.

Proof.

The key observation is that if without loss of generality $f$ has no zeroth Fourier modes, then the convolution integral

\nabla^{2}_{g_{a}}G_{a}*f=(\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a})*f,

but Lemma 3.18 says the integral kernel $\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}$ has exponential decay, at a rate faster than the exponential decay rate of $f$ itself. Thus at any point $p$ in the region $\{\ell>A^{1/2}\}\cap\mathcal{B}^{+}_{\nu}$ , the contribution to $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{p}$ from sources outside the ball $\{|p-q|_{a}^{\prime}\lesssim A^{1/2}\}$ is negligible. The contribution from sources inside the ball is treated by standard Schauder theory, and inherits the same exponential decay factor $e^{-\kappa\tilde{\ell}}$ as $f$ itself. ∎

Combining the Lemmas shows the main result of this Section after bootstrap.

Proposition 3.21.

(Periodic Euclidean region) In the situation of Lemma 3.19, in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu.

The constant only depends on $k,\alpha,\delta,\kappa$ and the scale-invariant uniform ellipticity bound on $a_{ij}$ .

We also record the following variant (cf. Section 3.4 for definition of norm).

Proposition 3.22.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ . Then in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta}}\leq C.

We do not need the extra log factor $\nu$ in the RHS because $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ implies power law decay on $f$ in the generic region, wheras $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ implies no decay.

Remark 3.8.

In the small ball $\{|\vec{\mu}|_{a}<A^{-1/4}\}$ the norms are not defined yet, but the regularity of $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is well controlled by $\Delta_{a}$ -harmonicity, since here $f=0$ by assumption.