16.14 Obstacles with bounded metric Laplacian74F4AD99CB8E

Proof. Fix $c>0$. By lemma 10.2.5(3), applied to functions, on which $\mathcal L=L$, the operator $P=L+c$ is a bijection of the space of distributions on $X$ onto itself, and its inverse maps $H^{-1}$ boundedly into $H^1$ and $H^0=L^2$ (lemma 10.2.1(2)) boundedly into $H^2$. Since no derivative of $f$ is known in advance, we gain regularity in two steps, using two cutoff functions.

For a smooth function $\chi $ on $X$, the Leibniz rules $\mathrm d(\chi u)=\chi \, \mathrm du+u\, \mathrm d\chi $ and $\mathrm d^*(\chi \alpha )=\chi \, \mathrm d^*\alpha -\langle \alpha ,\mathrm d\chi \rangle $, for smooth functions $u$ and real $1$-forms $\alpha $, give

where $\langle \cdot ,\cdot \rangle $ is the pointwise inner product of real $1$-forms, extended bilinearly. Since $L$ and multiplication by $\chi $ are formally self-adjoint, the transpose of $[L,\chi ]$ is $-[L,\chi ]$, and (16.14.5) extends to distributions $u$, with $\langle [L,\chi ]u,\phi \rangle =-\langle u,[L,\chi ]\phi \rangle $ for smooth $\phi $.

By proposition 5.2.6, choose smooth functions $\chi _0,\chi _1$ on $X$ with compact supports in $V$ such that $\chi _1=1$ on a neighborhood of $\overline{U'}$ and $\chi _0=1$ on a neighborhood of $\operatorname{Supp}\chi _1$. Extend $\chi _0f$ by zero to a function on $X$. By (16.14.5),

The first term lies in $L^2$, with norm at most $\| Lf\| _{L^2(V)}+c\| f\| _{L^2(V)}$, hence in $H^{-1}$ with a comparable norm. For smooth $\phi $, the function $[L,\chi _0]\phi $ is a first-order expression in $\phi $ with smooth coefficients supported in $V$, so

because the norm of $H^1$ dominates the $L^2$ norms of the first derivatives on compact subsets of the charts (lemma 10.2.1(2)). Hence $[L,\chi _0]f\in H^{-1}$, and the right side of the last identity has $H^{-1}$ norm at most $C(\| Lf\| _{L^2(V)}+\| f\| _{L^2(V)})$. Applying $P^{-1}$, we obtain $\chi _0f\in H^1$ with the same bound.

In particular $f$ has square-integrable first derivatives on the interior of $\{ \chi _0=1\} $, which contains $\operatorname{Supp}\chi _1$, with $L^2$ norm bounded in the same way. Therefore the distribution $[L,\chi _1]f=-2\langle \mathrm d\chi _1,\mathrm df\rangle +(L\chi _1)f$ is a square-integrable function, and in the identity

the right side lies in $L^2$, with norm at most $C(\| Lf\| _{L^2(V)}+\| f\| _{L^2(V)})$. Since $P^{-1}$ is bounded from $L^2$ to $H^2$, it follows that $\chi _1f\in H^2\subset W^{2,2}(X)$, with this bound for its norm in $W^{2,2}(X)$. Since $\chi _1=1$ on $U'$, this proves the estimate.

Finally, let $\psi \colon U_0\to \mathbb C^ n$ be a holomorphic chart with $U_0\subset U$, and let $U''\Subset U_0$ be open. Applying the preceding argument with $U'=U''$ and some open $V$ with $U''\Subset V\Subset U$, we find $\chi _1f\in W^{2,2}(X)$ with $\chi _1=1$ on $U''$, so $f\circ \psi ^{-1}\in W^{2,2}(\psi (U''))$ by (16.14.3). Hence $f\in W^{2,2}_{\mathrm{loc}}(U)$. The last assertion holds because continuous functions and locally bounded measurable functions are locally square-integrable. $\square$

Proof. The potentials $\rho _ j$ exist by lemma 6.14.2. By lemma 16.14.4, all $u_ j$ belong to $W^{2,2}_{\mathrm{loc}}(U)\subset W^{2,1}_{\mathrm{loc}}(U)$, so the matrices $A_ j$ and $B_ j$ are defined almost everywhere, and the positive current $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j=\mathrm{dd}^{\mathrm{c}}(\rho _ j+u_ j)$ has coefficients in $L^2_{\mathrm{loc}}$.

We first show that $A_ j$ is positive semidefinite almost everywhere. For each $\xi \in \mathbb C^ n$ the distribution $\sum _{a,b}(\rho _ j+u_ j)_{a\bar b}\xi _ a\bar\xi _ b$ is positive, because $T_ j$ is a positive current; since it is a locally integrable function, it is nonnegative almost everywhere, by proposition 2.5.3(3),(4) and lemma 2.5.1(2) (its mollifications are nonnegative and converge to it in $L^1_{\mathrm{loc}}$). We apply this to the countably many $\xi $ with rational real and imaginary parts. Outside the union of the corresponding null sets, $\sum _{a,b}(A_ j)_{ab}\xi _ a\bar\xi _ b\geq 0$ for all such $\xi $, hence for all $\xi \in \mathbb C^ n$, because these $\xi $ are dense and a quadratic form on $\mathbb C^ n$ is continuous.

Next we bound the coefficients. By the trace identity (10.2.13),

which is bounded on $W$, since $Lu_ j$ is locally bounded and $\overline W$ is compact. Since $T_ j=\frac{\mathrm i}{2\pi }\sum _{a,b}(A_ j)_{ab}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$, the function $\Lambda T_ j$ is the trace $\sum _{a,b}h^{a\bar b}(A_ j)_{ab}=\operatorname {tr}(H^{-1}A_ j)$ of $A_ j$ relative to $h$, as in (10.2.14), where $H=(h_{a\bar b})$. As $\overline W$ is compact, $H\leq CI$ on $W$ for a constant $C$, so $H^{-1}\geq C^{-1}I$; since $A_ j$ is positive semidefinite, $\operatorname {tr}(H^{-1}A_ j)=\operatorname {tr}(A_ j^{1/2}H^{-1}A_ j^{1/2})\geq C^{-1}\operatorname {tr}A_ j$ almost everywhere on $W$. For a positive semidefinite Hermitian matrix every diagonal entry is at most the trace, and every entry satisfies $|a_{ab}|^2\leq a_{aa}a_{bb}$, by the positivity of the principal $2\times 2$ minors. Therefore $A_ j$ is bounded on $W$.

The functions $\rho _ j+u_ j$ are continuous and plurisubharmonic on $W$, and they have bounded complex Hessian in the sense of section 7.13. The first identity is therefore the first part of proposition 7.13.4. The functions $\rho _ j+f_ j$ are real Borel functions in $W^{2,1}_{\mathrm{loc}}(W)$, and adding $\rho _ j$ to both $u_ j$ and $f_ j$ does not change the contact set $\{ u_ j=f_ j\} $. Hence the second part of proposition 7.13.4 gives the second identity. Moreover proposition 7.13.3, applied to $\rho _ j+u_ j$ and $\rho _ j+f_ j$, shows that $A_ j=B_ j$ almost everywhere on $\{ u_ j=f_ j\} $, hence on $E$. Consequently $\mathbf1_ ED(B_1,\ldots ,B_ n)=\mathbf1_ ED(A_1,\ldots ,A_ n)$ almost everywhere. This density is nonnegative, since the mixed discriminant of positive semidefinite matrices is nonnegative (lemma 5.3.3(3)), and it is bounded, since $D$ is a polynomial in the entries of the bounded matrices $A_ j$. $\square$

Proof. First let $A_0$ be a constant real symmetric matrix with $A_0\geq \lambda I$, and put $M=A_0^{-1/2}$, a symmetric matrix of operator norm at most $\lambda ^{-1/2}$. For $v\in C_ c^\infty (\mathbb R^ d)$ let $\tilde v(y)=v(M^{-1}y)$, so that $v(x)=\tilde v(Mx)$. Then $D^2v(x)=M\, D^2\tilde v(Mx)\, M$ and $A_0:D^2v(x)=\operatorname {tr}(MA_0M\, D^2\tilde v(Mx))=\Delta \tilde v(Mx)$. Since $|MNM|\leq \lambda ^{-1}|N|$ for every matrix $N$, the change of variables $y=Mx$ and (2.9.13) give

This estimate extends to every $w\in W^{2,p}(\mathbb R^ d)$ that vanishes almost everywhere outside a compact set $K$. Indeed, for $0<\varepsilon \leq 1$ the mollifications $w*\rho _\varepsilon $ are test functions supported in the compact set $K'=\{ x:\operatorname {dist}(x,K)\leq 1\} $, their second derivatives are $(\partial _\mu \partial _\sigma w)*\rho _\varepsilon $, by proposition 2.5.3(1), and these converge to $\partial _\mu \partial _\sigma w$ in $L^ p(K')$, by lemma 2.5.1(2), hence in $L^ p(\mathbb R^ d)$, since all these functions vanish almost everywhere outside $K'$. Applying the estimate to $v=w*\rho _\varepsilon $ and letting $\varepsilon \to 0$, we obtain it for $w$.

Now let $A$ and $w$ be as in the statement, put $C_1=\lambda ^{-1}C_{d,p}$, $\delta _0=1/(2C_1)$, and $A_0=A(x_ B)$. Since $|\sum _{\mu ,\sigma }m_{\mu \sigma }m'_{\mu \sigma }|\leq |m|\, |m'|$ for matrices $m$ and $m'$, we have $|A_0:D^2w-A:D^2w|\leq \delta _0|D^2w|$ almost everywhere on $B$, and both sides vanish outside $B$. Hence

and since $\| D^2w\| _ p<\infty $ we obtain the lemma with $C=2C_1$. $\square$

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. $\square$

Proof. By proposition 16.14.9 and (16.14.3), every $f_ j$ belongs to $W^{2,p}_{\mathrm{loc}}$ in every chart, for every finite $p$. In a chart, the current $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j$ is therefore given by integration against the form $\theta _ j+\frac{\mathrm i}{2\pi }\sum _{a,b}(f_ j)_{a\bar b}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$, where the $(f_ j)_{a\bar b}$ are the weak derivatives, as in (5.2.9); its coefficients lie in $L^ p_{\mathrm{loc}}$. On the overlap of two charts, the forms obtained in the two charts represent the same current. For a current with locally integrable coefficients, (5.3.6), in which $\delta $ is the Jacobian determinant relating the two Lebesgue measures, says that the coefficients in the two charts are related, as distributions and hence almost everywhere (proposition 2.5.3(4)), by the pointwise change of variables of forms. Thus the two forms agree almost everywhere as pointwise forms. Hence the pointwise wedge product of the forms $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j$ is a well-defined $(n,n)$-form, almost everywhere on $X$, and we write it as $F\, \mathrm dV$.

A coefficient of the wedge product in a chart is a sum of products of $n$ coefficients of the factors and of smooth functions; by Hölder's inequality, with each factor in $L^{nq}$ on compact subsets of the chart, these products lie in $L^ q$ there, for every $1\leq q<\infty $. Covering $X$ by finitely many compact subsets of charts, we obtain $F\in L^ q(X)$. A real function in $L^1(X)$ defines a finite signed Radon measure through its positive and negative parts.

On a coordinate ball, $\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j=\mathrm{dd}^{\mathrm{c}}(\rho _ j+f_ j)=\frac{\mathrm i}{2\pi }\sum _{a,b}(B_ j)_{ab}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$ almost everywhere, and the identity $\omega _{A_1}\wedge \cdots \wedge \omega _{A_ n}=D(A_1,\ldots ,A_ n)\, \omega _ I^ n$ of lemma 5.3.3(2), applied pointwise with $A_ j=B_ j/(2\pi )$, together with $\omega _ I^ n=n!\, 2^ n\, \mathrm dV_{2n}$, gives $\bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}f_ j)=n!\pi ^{-n}D(B_1,\ldots ,B_ n)\, \mathrm dV_{2n}$. Finally, on every coordinate ball $W$ the second identity of proposition 16.14.6, applied with $U=X$, shows that $\mathbf1_ E\bigwedge _ j(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)=\mathbf1_ EF\, \mathrm dV$ on $W$. Since coordinate balls cover $X$, the two measures agree on $X$. $\square$