Lemma 16.14.475E3F7AC3FA0

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. $\square$