Proposition 16.14.9F9586D89B24F

Proof. We first treat smooth functions and then pass to the general case through the inverse of $L+c$.

Step 1: the estimate for smooth real functions. Let $d=2n$. Fix finitely many holomorphic charts and compact subsets $K_\iota $ of their domains whose interiors cover $X$. By (16.14.8), in these charts $L=-4\pi \, A:D^2$ on real functions, with smooth real symmetric matrices $A$; by compactness there is $\lambda >0$ with $A\geq \lambda I$ on every $K_\iota $. Let $\delta _0$ and $C$ be the constants of lemma 16.14.7 for $d$, $p$ and $\lambda $. Since $A$ is uniformly continuous on a compact neighborhood of each $K_\iota $, there is $r>0$ such that, for every $x\in K_\iota $, the ball $B(x,r)$ lies in that neighborhood and $|A(y)-A(x)|\leq \delta _0$ for $y\in B(x,r)$. Cover $X$ by finitely many such balls $B_\nu $, and take a smooth partition of unity $(\chi _\nu )$ with $\operatorname{Supp}\chi _\nu $ a compact subset of $B_\nu $ (proposition 5.2.6).

Let $f$ be smooth and real, and let $v_\nu =\chi _\nu f$, written in the chart of $B_\nu $ and extended by zero. By lemma 16.14.7, $\| D^2v_\nu \| _ p\leq C\| A:D^2v_\nu \| _ p=(C/4\pi )\| Lv_\nu \| _ p$. By (16.14.5), $Lv_\nu =\chi _\nu Lf+[L,\chi _\nu ]f$, where the commutator involves only $f$ and its first derivatives. Summation therefore gives

Here and below the $L^ p$ norms of $f$ and $\nabla f$ on $X$ may be computed in the charts or with the metric of $\omega $; on the fixed finite cover these choices give equivalent norms. Since $f=\sum _\nu v_\nu $, the norm $\| \nabla f\| _ p$ is at most a fixed multiple of $\sum _\nu \| \nabla v_\nu \| _ p$. By lemma 2.9.15, for every $\varepsilon >0$,

Combining the last three estimates and choosing $\varepsilon $ small, we can absorb $S_2$, which is finite because $f$ is smooth. This gives $S_2\leq C(\| Lf\| _ p+\| f\| _ p)$ and then $\| \nabla f\| _ p\leq C(\| Lf\| _ p+\| f\| _ p)$. Since the norm of $W^{2,p}(X)$ may be computed, up to equivalence, with the charts of the balls $B_\nu $ and the partition of unity $(\chi _\nu )$ (subsection 2.9.9), this proves (16.14.10) for smooth real $f$.

Step 2: an $L^ p$ bound for the inverse. Fix $c>0$ and put $P=L+c$. For a smooth real function $\psi $, lemma 10.2.5(3) gives a unique distributional solution $u=P^{-1}\psi $, which lies in every $H^ r$ and is therefore smooth, by lemma 10.2.1(3); it is real since $P$ is a real operator. For $\eta >0$ put

The function $\phi _\eta $ is smooth, so we may multiply the equation $Lu+cu=\psi $ by $\phi _\eta (u)$ and integrate. Since $\mathrm d^*$ is the formal adjoint of $\mathrm d$, we have $\int _ X(Lu)\phi _\eta (u)\, \mathrm dV=\int _ X\phi _\eta '(u)|\mathrm du|^2\, \mathrm dV\geq 0$, and therefore

As $\eta \downarrow 0$, $u\phi _\eta (u)\to |u|^ p$ and $\phi _\eta (u)\to u|u|^{p-2}$ pointwise, the latter being $0$ where $u=0$. For $p\geq 2$ and $\eta \leq 1$ both functions are bounded by $(1+u^2)^{p/2}$, which is bounded since $u$ is continuous on the compact manifold $X$; for $1<p<2$ we have $(u^2+\eta )^{(p-2)/2}\leq |u|^{p-2}$ where $u\neq 0$, so they are bounded by $|u|^ p$ and $|u|^{p-1}$. Dominated convergence and Hölder's inequality therefore give $c\| u\| _ p^ p\leq \| \psi \| _ p\| u\| _ p^{p-1}$, that is (if $u\neq 0$; otherwise there is nothing to prove),

Applying Step 1 to $u$, for which $Lu=\psi -cu$, we conclude that $\| P^{-1}\psi \| _{W^{2,p}}\leq C\| \psi \| _ p$.

Step 3: the general case. Let $f$ be real, put $\psi =Lf+cf\in L^ p(X)$, and choose smooth real functions $\psi _ j\to \psi $ in $L^ p(X)$, using chart cutoffs and mollification (lemma 2.5.1). By Step 2 the solutions $u_ j=P^{-1}\psi _ j$ form a Cauchy sequence in $W^{2,p}(X)$. This space is complete: in each chart the derivatives of order at most two converge in $L^ p$, and testing against smooth functions identifies the limits with the weak derivatives of the limit function. Let $u$ be the limit. Then $Pu_ j\to Pu$ in the sense of distributions, so $Pu=\psi $. Thus $P(u-f)=0$, and since $P$ is injective on distributions (lemma 10.2.5(3)), $u=f$. Passing to the limit in the estimate of Step 2 gives $\| f\| _{W^{2,p}}\leq C\| \psi \| _ p\leq C(\| Lf\| _ p+\| f\| _ p)$. For complex $f$ we apply this to the real and imaginary parts, since $L$ is a real operator.

Step 4: continuous functions with bounded Laplacian. If $f$ is continuous and $Lf$ is bounded, then $f$ and $Lf$ lie in every $L^ p(X)$ with $p<\infty $, which proves the assertion for $1<p<\infty $; the case $p=1$ follows from the case $p=2$, since $X$ has finite volume. Given $0<\alpha <1$, choose $p>d/(1-\alpha )$. In every chart, $f$ belongs to $W^{2,p}_{\mathrm{loc}}$ by (16.14.3), so Morrey's inequality (lemma 2.9.16) gives a representative of $f$ in $C^{1,1-d/p}_{\mathrm{loc}}$, hence in $C^{1,\alpha }_{\mathrm{loc}}$. This representative agrees with the continuous function $f$ almost everywhere, hence everywhere. $\square$