Lemma 10.2.1D02C1B7CE2FA

Proof. In this proof, the constants $C$ depend on $r$, $l$ and the data indicated, but not on the forms or functions involved. We first compare the coefficient vectors of a form in two charts, and then prove (1)–(5) in turn.

(a) A change of charts. Let $\kappa \colon U\to \mathbb R^{2n}$ and $\kappa '\colon U'\to \mathbb R^{2n}$ be charts of $X$ with frames of the bundle of forms over $U$ and $U'$, let $K\subset U$ be compact, and let $W\subset U'$ be an open set whose closure is a compact subset of $U'$. For a smooth form $w$ that vanishes outside $K$, let $w_\kappa $ and $w_{\kappa '}$ be its coefficient vectors in the two charts and frames, the first one extended by zero to $\mathbb R^{2n}$. We claim that

with $C$ depending on the charts, frames, $K$ and $W$. Indeed, on $\kappa '(U\cap U')$ we have $w_{\kappa '}(y)=A(y)\, w_\kappa (\tau (y))$, where $\tau =\kappa \circ \kappa '^{-1}$ and $A$ is the smooth matrix of the change of frames. By the chain rule, the Leibniz rule and induction on $|\alpha |$,

where the matrices $c_{\alpha \beta }$ are polynomials in the derivatives of order at most $|\alpha |$ of $A$ and of $\tau $; they are smooth on $\kappa '(U\cap U')$ and therefore bounded on its compact subset $\kappa '(\overline W\cap K)$. Since $w$ vanishes on the open set $X\setminus K$, all derivatives of $w_{\kappa '}$ vanish on $\kappa '(W\setminus K)$, and every point of $\kappa '(W)$ lies in this set or in $\kappa '(W\cap K)$. Hence, for $|\alpha |\leq r$,

by the change of variables $x=\tau (y)$, whose Jacobian determinant $|\det D\tau ^{-1}|$ is bounded on the compact set $\kappa (\overline W\cap K)$. This proves (10.2.2). We also use the Leibniz rule in the following form. If $M$ is a smooth matrix-valued function on an open set $V\subset \mathbb R^{2n}$ whose derivatives of order at most $r$ are bounded on $V$, then $\| Mv\| _{(r),V}\leq C\| v\| _{(r),V}$ for smooth $v$ on $V$, with $C$ depending on these bounds. If $v$ vanishes outside a compact set $L\subset V$, it suffices that $M$ be smooth on $V$, the bounds being taken on $L$, and then $\| Mv\| _{(r)}\leq C\| v\| _{(r)}$ for the extensions by zero.

(b) Proof of (1). For a test function $v$ on $\mathbb R^{2n}$ we have $\widehat{\partial ^\beta v}(\xi )=(2\pi \mathrm i\xi )^\beta \widehat v(\xi )$, and $(1+|\xi |^2)^ r=\sum _{|\beta |\leq r}c_\beta |\xi ^\beta |^2$ with positive multinomial coefficients $c_\beta $. Hence Plancherel's theorem (theorem 2.9.4) gives

and the norm of $H^ r$ is equivalent on smooth forms to $\bigl (\sum _ k\| \chi _ ku\| _{(r)}^2\bigr )^{1/2}$. Let charts with domains $U'_ m$, frames over them, and cutoffs $\chi '_ m$ with $\sum _ m\chi '_ m=1$ be other choices, and choose open sets $W_ m$ with $\operatorname{Supp}\chi '_ m\subset W_ m$ and $\overline{W_ m}$ compact in $U'_ m$. For a smooth form $u$ we have $\chi '_ mu=\sum _ k\chi '_ m\chi _ ku$, and the form $\chi '_ m\chi _ ku$ vanishes outside the compact set $\operatorname{Supp}\chi _ k$ and outside $\operatorname{Supp}\chi '_ m\subset W_ m$. By (10.2.2), with $K=\operatorname{Supp}\chi _ k$ and $W=W_ m$, its norm $\| \cdot \| _{(r)}$ in the $m$th new chart is at most $C$ times its norm in the $k$th old chart, and by the Leibniz rule, applied to the factor $\chi '_ m$ on the compact set $\kappa _ k(\operatorname{Supp}\chi _ k)$, where $\kappa _ k$ is the $k$th chart, the latter is at most $C\| \chi _ ku\| _{(r)}$. Summing over $k$ and $m$, we see that the new norm is bounded by a constant times the old one, and by symmetry the two norms are equivalent.

If $M$ is a smooth function or endomorphism, then $\chi _ kMu=M\chi _ ku$ vanishes outside $\operatorname{Supp}\chi _ k$, so the Leibniz rule gives $\| \chi _ kMu\| _{(r)}\leq C\| \chi _ ku\| _{(r)}$. Finally, $\| \chi _ k\phi \| _{(r)}$ is bounded, by the Leibniz rule and since $\kappa _ k(\operatorname{Supp}\chi _ k)$ has finite measure, by a constant times the suprema on $\operatorname{Supp}\chi _ k$ of the derivatives of order at most $r$ of the coefficients of $\phi $; hence $\| \phi \| _{H^ r}\leq C\| \phi \| _{C^ r}$.

(c) Proof of (2). For a smooth form $u$ we have $|u|\leq \sum _ k|\chi _ ku|$ pointwise, since $\chi _ k\geq 0$ and $\sum _ k\chi _ k=1$. On the compact sets $\operatorname{Supp}\chi _ k$ the pointwise norm of forms is comparable with the Euclidean norm of the coefficient vectors, and $\mathrm dV$ with Lebesgue measure in the charts; hence $\| u\| _{L^2}\leq C\sum _ k\| \chi _ ku\| _{(0)}\leq C\| u\| _{H^ r}$, and conversely $\| \chi _ ku\| _{(0)}\leq C\| u\| _{L^2}$ because $0\leq \chi _ k\leq 1$. Therefore the identity map of smooth forms extends to a bounded map $j$ from $H^ r$ to the space of $L^2$ forms. Let $u\in H^ r$ be the limit in $H^ r$ of smooth forms $u_ i$. By (1), for every $k$ and $|\alpha |\leq r$ the functions $\partial ^\alpha (\chi _ ku_ i)$ form a Cauchy sequence in $L^2(\mathbb R^{2n})$, with a limit $g_{k,\alpha }$, while $\chi _ ku_ i\to \chi _ kj(u)$ in $L^2$. For every test function $\phi $ on $\mathbb R^{2n}$,

so $g_{k,\alpha }$ is the distributional derivative $\partial ^\alpha (\chi _ kj(u))$. Hence these derivatives are square integrable, and $\sum _ k\| \chi _ kj(u)\| _{(r)}^2=\lim _ i\sum _ k\| \chi _ ku_ i\| _{(r)}^2$, which by (1) lies between two positive multiples of $\lim _ i\| u_ i\| _{H^ r}^2=\| u\| _{H^ r}^2$. In particular, $j(u)=0$ implies $u=0$, so $j$ is injective, and we identify $u$ with $j(u)$. This proves the first assertion, the necessity of the condition and the equivalence of the norms.

Conversely, let $u$ be an $L^2$ form such that every $\chi _ ku$ has square integrable distributional derivatives of order at most $r$, and let $\rho _\varepsilon $ be the mollifier of section 2.5. Since $\chi _ ku$ vanishes outside the compact set $\kappa _ k(\operatorname{Supp}\chi _ k)$, the functions $v_{k,\varepsilon }=\rho _\varepsilon *(\chi _ ku)$ are smooth and, for small $\varepsilon $, vanish outside a fixed compact subset $L_ k$ of $\kappa _ k(U_ k)$, where $U_ k$ is the domain of the $k$th chart; moreover $\partial ^\alpha v_{k,\varepsilon }=\rho _\varepsilon *\partial ^\alpha (\chi _ ku)$ by proposition 2.5.3, and this converges to $\partial ^\alpha (\chi _ ku)$ in $L^2$ as $\varepsilon \to 0$, by lemma 2.5.1, for $|\alpha |\leq r$. Let $w_{k,\varepsilon }$ be the smooth form on $X$ whose coefficient vector in the $k$th chart and frame is $v_{k,\varepsilon }$, extended by zero outside $U_ k$, and put $u_\varepsilon =\sum _ kw_{k,\varepsilon }$. Choose open sets $W_ m$ with $\operatorname{Supp}\chi _ m\subset W_ m$ and $\overline{W_ m}$ compact in $U_ m$. By (10.2.2), with $K=\kappa _ k^{-1}(L_ k)$ and $W=W_ m$, and by the Leibniz rule for the factor $\chi _ m$, we have $\| \chi _ m(w_{k,\varepsilon }-w_{k,\varepsilon '})\| _{(r)}\leq C\| v_{k,\varepsilon }-v_{k,\varepsilon '}\| _{(r)}$. By (1), $(u_\varepsilon )$ is therefore a Cauchy sequence in $H^ r$ as $\varepsilon \to 0$ along a sequence, and it converges in $L^2$ to $\sum _ k\chi _ ku=u$, by the comparison of pointwise norms on the compact sets $\kappa _ k^{-1}(L_ k)\cup \operatorname{Supp}\chi _ k\subset U_ k$. Hence $u\in H^ r$.

For $r=0$ the condition holds for every $L^2$ form, so $H^0$ is the space of all $L^2$ forms, and its norm is equivalent to the $L^2$ norm by the first paragraph. Other choices of the charts, frames and cutoffs give, by (1), equivalent norms on smooth forms, hence the same Cauchy sequences and the same limits in $L^2$; so they give the same subspace $H^ r$ with an equivalent norm. If $r\leq r'$, then $\| \phi \| _{H^ r}\leq \| \phi \| _{H^{r'}}$ for smooth $\phi $, so every Cauchy sequence in $H^{r'}$ is a Cauchy sequence in $H^ r$ with the same limit in $L^2$; hence $H^{r'}\subset H^ r$.

Finally, let $W$ be open with closure compact in a chart with a frame. For smooth $u$, we have $u=\sum _ m\chi _ mu$, and (10.2.2), applied to $w=\chi _ mu$ with $K=\operatorname{Supp}\chi _ m$, gives $\| u\| _{(r),W}\leq C\sum _ m\| \chi _ mu\| _{(r)}\leq C\| u\| _{H^ r}$. If $u\in H^ r$ is the limit in $H^ r$ of smooth forms $u_ i$, the coefficient vectors of the $u_ i$ on $W$ form a Cauchy sequence for $\| \cdot \| _{(r),W}$ and converge in $L^2(W)$ to that of $u$. Testing against test functions on $W$ as above, we find that their derivatives of order at most $r$ converge in $L^2(W)$ to those of the coefficient vector of $u$, and the bound passes to the limit.

(d) Proof of (3). Let first $v$ be a smooth $\mathbb C^ N$-valued function with compact support, and let $|\alpha |\leq l$. The inversion formula of theorem 2.9.4, applied to $\partial ^\alpha v$, whose Fourier transform is $(2\pi \mathrm i\xi )^\alpha \widehat v$, and the Cauchy–Schwarz inequality give

where the first integral on the right is finite, by polar coordinates, because $2(r-l)>2n$. By (10.2.3), $\sup |\partial ^\alpha v|\leq C\| v\| _{(r)}$. If $v$ is only compactly supported with $\| v\| _{(r)}<\infty $, then, as in (c), the mollifications $v_\varepsilon =\rho _\varepsilon *v$ are smooth with compact support and $\| v_\varepsilon -v\| _{(r)}\to 0$ as $\varepsilon \to 0$. Along a sequence $\varepsilon _ i\to 0$, the estimate shows that the derivatives of order at most $l$ of the $v_{\varepsilon _ i}$ converge uniformly; hence the $v_{\varepsilon _ i}$ converge uniformly to a function of class $C^ l$ whose derivatives of order at most $l$ are bounded by $C\| v\| _{(r)}$. This function agrees with $v$ almost everywhere, since $v_{\varepsilon _ i}\to v$ in $L^2$ and a subsequence converges almost everywhere.

Now let $u$ be a smooth form. On $\operatorname{Supp}\chi _ k$, the coefficient vector of $u$ in the $k$th chart and frame is the sum over $m$ of the coefficient vectors of the forms $\chi _ mu$. We apply the formula for $\partial ^\alpha w_{\kappa '}$ in (a) with $\kappa $ the $m$th chart, $\kappa '$ the $k$th chart and $K=\operatorname{Supp}\chi _ m$. The derivatives of the coefficient vector of $\chi _ mu$ in the $k$th chart vanish outside $\kappa _ k(\operatorname{Supp}\chi _ m)$, and the coefficients $c_{\alpha \beta }$ are bounded on the compact set $\kappa _ k(\operatorname{Supp}\chi _ k\cap \operatorname{Supp}\chi _ m)$. Hence the derivatives of order at most $l$ of this coefficient vector are bounded on $\kappa _ k(\operatorname{Supp}\chi _ k)$ by $C$ times the suprema of the derivatives of order at most $l$ of the coefficient vector of $\chi _ mu$ in the $m$th chart, and the Euclidean estimate and (1) give $\| u\| _{C^ l}\leq C\sum _ m\| \chi _ mu\| _{(r)}\leq C\| u\| _{H^ r}$.

If $u\in H^ r$ is the limit in $H^ r$ of smooth forms $u_ i$, then $\| u_ i-u_{i'}\| _{C^ l}\leq C\| u_ i-u_{i'}\| _{H^ r}$, so in every chart the coefficients of the $u_ i$ and their derivatives of order at most $l$ converge uniformly on $\operatorname{Supp}\chi _ k$, and hence on the open sets $\{ \chi _ k>0\} $, which cover $X$. The limit is a form of class $C^ l$ with $\| \cdot \| _{C^ l}\leq C\| u\| _{H^ r}$, and it agrees with $u$ almost everywhere, because $u_ i\to u$ in $L^2$. Finally, if $u$ lies in every $H^ r$, then for each $l$ it agrees almost everywhere with a form of class $C^ l$; these continuous forms agree almost everywhere, hence everywhere, so $u$ agrees almost everywhere with a smooth form.

(e) Proof of (4). Let $u\in H^{-r}$. Since smooth forms are dense in $H^ r$, the bounded functional $\phi \mapsto \langle u,\phi \rangle $ on smooth forms extends uniquely to a bounded conjugate-linear functional on $H^ r$, with the same norm. Conversely, let $\ell $ be a bounded conjugate-linear functional on $H^ r$. Since $*$ is a real bundle isomorphism with smooth coefficients, the formula $T(\alpha )=\ell (\overline{*^{-1}\alpha })$ defines a complex-linear functional $T$ on smooth forms. If $\alpha $ is supported in a compact subset $K$ of a chart, then, by (1) and the chain rule on the compact sets $K\cap \operatorname{Supp}\chi _ k$, we have $|T(\alpha )|\leq \| \ell \| \, \| *^{-1}\overline\alpha \| _{H^ r}\leq C\| *^{-1}\overline\alpha \| _{C^ r}\leq C_ K\sum _ Jp_{K,r}(\alpha _ J)$ in the notation of section 5.3. Hence the components of $T$ of the various degrees are currents, and $T$ is a distributional form $u$ with $\langle u,\phi \rangle =u(*\overline\phi )=\ell (\phi )$ for smooth $\phi $. Thus $u\in H^{-r}$ and $\| u\| _{H^{-r}}=\| \ell \| $. It is unique, because $*$ maps the smooth forms onto the smooth forms, so that a distributional form is determined by its pairings with smooth forms.

For a smooth function or endomorphism $M$, with pointwise adjoint $M^\dagger $, we have $\langle Mu,\phi \rangle =\langle u,M^\dagger \phi \rangle $ for distributional forms $u$ and smooth $\phi $, so $|\langle Mu,\phi \rangle |\leq \| u\| _{H^{-r}}\| M^\dagger \phi \| _{H^ r}\leq C\| u\| _{H^{-r}}\| \phi \| _{H^ r}$ by (1). The inclusion $H^{-r}\subset H^{-r'}$ for $r\leq r'$ follows from $\| \phi \| _{H^ r}\leq \| \phi \| _{H^{r'}}$, and by (1) other choices of the charts, frames and cutoffs change the norm of $H^ r$ on smooth forms, and hence $H^{-r}$, only up to equivalence.

The definition of $H^ r$, $r\geq 0$, does not involve $\omega $. Let $\omega '$ be another Kähler form, with pointwise inner product $\langle \cdot ,\cdot \rangle '$ and volume form $\mathrm dV'$. At each point, the Hermitian form $\langle a,b\rangle '\, \mathrm dV'/\mathrm dV$ on forms is $\langle a,Sb\rangle $ for a unique endomorphism $S$, which is positive definite, hence invertible, and depends smoothly on the point. For smooth $u$ and $\phi $ we then have $u\wedge *'\overline\phi =\langle u,\phi \rangle '\, \mathrm dV'=\langle u,S\phi \rangle \, \mathrm dV=u\wedge *\overline{S\phi }$, where $*'$ is the Hodge star of $\omega '$; since $u$ is arbitrary, $*'\overline\phi =*\overline{S\phi }$. Hence the pairing defined by $\omega '$ is $u(*'\overline\phi )=\langle u,S\phi \rangle $ for every distributional form $u$. Since multiplication by $S$ and by $S^{-1}$ is bounded on $H^ r$ by (1), the condition defining $H^{-r}$ is the same for both pairings, and the two norms are equivalent.

(f) Proof of (5). Let $u$ be a distributional form. Since $\sum _ k\chi _ k=1$, we have $\langle u,\phi \rangle =\sum _ ku(\chi _ k\, {*\overline\phi })$ for smooth $\phi $. The forms $\chi _ k\, {*\overline\phi }$ are supported in the compact sets $\operatorname{Supp}\chi _ k$ of the charts, so the definition of currents (section 5.3) gives integers $m_ k$ and constants $C_ k$ with $|u(\chi _ k\, {*\overline\phi })|\leq C_ k\sum _ Jp_{\operatorname{Supp}\chi _ k,m_ k}\bigl ((\chi _ k\, {*\overline\phi })_ J\bigr )$. By the Leibniz rule, and since $*$ and the change from the coordinate frames to the given frames have smooth coefficients, the right side is at most $C\| \phi \| _{C^ l}$ with $l=\max _ km_ k$. By (3), $\| \phi \| _{C^ l}\leq C\| \phi \| _{H^ r}$ for $r>l+n$, and therefore $u\in H^{-r}$. $\square$