Compact Laplace-type operators and harmonic representatives

In dimension zero each chart is a point, the manifold is discrete, and a compact support is finite. A differential operator is fibrewise multiplication by its order-zero coefficient, so its fibre adjoint satisfies the required identity on each point and hence on every finite sum. Uniqueness and all three consequences are the fibre adjoint identities. For the remaining argument assume $n\ge 1$.

On a coordinate open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, let $a,b$ be compactly supported smooth scalar functions and let $\gamma >0$ be the smooth density coefficient. The product $a\bar{b}\gamma$, extended by zero, is smooth and compactly supported on ${\mathbb{R}}^n$. Apply [F1] to its derivative and [F2] to the product to obtain ${\int }_{\mathrm{\Omega }}({\mathrm{\partial }}_{i}a)\bar{b}\gamma dx=-{\int }_{\mathrm{\Omega }}a{\mathrm{\partial }}_i(\bar{b}\gamma )dx$. Since $\gamma$ is real, this is ${\int }_{\mathrm{\Omega }}({\mathrm{\partial }}_{i}a)\bar{b}\gamma dx={\int }_{\mathrm{\Omega }}a\overline{(-{\gamma }^{-1}{\mathrm{\partial }}_i(\gamma b))}\gamma dx$. The real case omits conjugations. Thus integration against the density transposes ${\mathrm{\partial }}_i$ to $-{\gamma }^{-1}{\mathrm{\partial }}_i\circ \gamma$.

Iterating step 1.2 and inserting the matrix coefficients gives the local formula of the Statement, so $P^{\star }$ exists locally. For a multi-index $\alpha$ and a matrix coefficient $a=a(x)$, the transpose of $a{D}^{\alpha }$ against the fibre pairing is the operator $v\mapsto (-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\gamma }^{-1}{D}^{\alpha }(\gamma a^{\ast }v)$: each single derivative contributes the factor $-{\gamma }^{-1}\circ {\mathrm{\partial }}_i\circ \gamma$, the matrix acts by the fibre adjoint $a^{\ast }$, and [F2] expands $D^{\alpha }(\gamma a^{\ast }v)$ with top term $\gamma a^{\ast }{D}^{\alpha }v$ and lower-order terms, exactly as in the displayed formula. Summing over $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$ gives a smooth operator of order at most two on the chart.

Vanishing on a chart. Suppose $\int ⟨u,Qv⟩d\mu =0$ for all compactly supported smooth tests $u,v$ in a chart. Fix $v$. If a component $(Qv{)}_{\lambda }$ is nonzero at $p$, choose a relatively compact neighbourhood of $p$ inside the chart where that component is nonzero. By [F5] choose a nonnegative smooth cutoff $\chi$ supported there and equal to one near $p$, and set $u=\chi (Qv{)}_{\lambda }{e}_{\lambda }$ in an orthonormal frame. The first-variable-linear convention gives $⟨u,Qv⟩=\chi \mathrm{\mid }(Qv{)}_{\lambda }{\mathrm{\mid }}^2$, whose density integral is positive, a contradiction. Thus $Qv=0$ for every compactly supported $v$. For an arbitrary smooth local $v$, a cutoff equal to one near any given point reduces to such a test; locality of a differential operator then gives $Qv=0$ at that point. Hence $Q=0$, and two local formal adjoints agree on overlaps.

Leading coefficients. In a chart with orthonormal frames, write the local coefficients of $P$ as $a_{\alpha }$, $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$. By [F2] the coefficient of $D^{\alpha }v$ in ${\gamma }^{-1}{D}^{\alpha }(\gamma a_{\alpha }^{\ast }v)$ is $a_{\alpha }^{\ast }$, and all other terms contributed by ${\gamma }^{-1}{D}^{\alpha }(\gamma \cdot )$ have order at most $\mathrm{\mid }\alpha \mathrm{\mid }-1$; terms with $\mathrm{\mid }\alpha \mathrm{\mid }<2$ have order at most one. Hence the order-$\delta$ leading coefficients of $P^{\star }$ are $(-1{)}^{\delta }{a}_{\alpha }^{\ast }$.

Gluing. The local operators of step 2.1 on the members of a chart cover whose frames are orthonormal by [F4] agree on every overlap by step 3.1, so they patch to one global smooth linear operator $P^{\star }:\mathrm{\Gamma }(F)\to \mathrm{\Gamma }(E)$ of order at most two; on a chart, $P^{\star }$ is the local construction there.

Global adjoint identity. Put $K=\mathrm{supp}u$ for compactly supported smooth $u,v$. If $K$ is empty the identity is zero. Otherwise, coordinate balls with compact closures inside orthonormal-frame charts cover $K$; compactness gives finitely many such balls $V_i$ in charts $U_i$. By [F5] choose nonnegative bumps ${\beta }_i$ supported in $U_i$ and equal to one near ${\bar{V}}_i$. Their sum $s$ is positive near $K$. Define $u_i=({\beta }_i\mathrm{/}s)u$ on $\{s>0\}$ and zero elsewhere. This extension is smooth because $u$ vanishes near every point of $\{s=0\}$; its support lies in the compact set $K\cap \mathrm{supp}{\beta }_i\subset U_i$, and $u={\sum }_i{u}_i$. Choose by [F5] a cutoff ${\theta }_i$ supported in $U_i$ and equal to one near $\mathrm{supp}{u}_i$, taking its support compact by working inside a relatively compact neighbourhood of that support in $U_i$. Locality gives $⟨P{u}_i,v⟩=⟨P{u}_i,{\theta }_{i}v⟩$ and $⟨u_i,P^{\star }({\theta }_{i}v)⟩=⟨u_i,P^{\star }v⟩$, since the relevant jets agree near $\mathrm{supp}{u}_i$. The local identity of step 2.1 therefore yields $\int ⟨P{u}_i,v⟩d\mu =\int ⟨u_i,P^{\star }v⟩d\mu$. Summing the finite family proves the global identity. Only finite selections are made here; Countable Choice is inherited by the cited density-integral convention.

Uniqueness. If a smooth operator $Q$ also satisfied the global adjoint identity, then $\int ⟨u,(P^{\star }-Q)v⟩d\mu =0$ for all compactly supported smooth $u,v$, so step 3.1 applied to $P^{\star }-Q$ on any chart gives $P^{\star }-Q=0$. Thus $P^{\star }$ is the unique such operator.

Consequences. For a Laplace-type $P$ the order-two coefficients are $a_{(i,j)}=-a^{ij}{\delta }_{\lambda \mu }$ with $(a^{ij})$ smooth symmetric positive definite, so those of $P^{\star }$ are $(-1{)}^2{a}_{(i,j)}^{\ast }=-a^{ij}{\delta }_{\lambda \mu }$, the same matrix; hence $P^{\star }$ is Laplace type and ${\sigma }_2(P^{\star })={\sigma }_2(P)$. Applying the defining identity twice and using step 6.1, $(P^{\star }{)}^{\star }=P$. Finally, with $E=F$, the difference of the defining identities for $P$ and $P^{\star }$ shows that $P$ is formally symmetric exactly when $\int ⟨s,(P-P^{\star })t⟩d\mu =0$ for all compactly supported smooth $s,t$, which by step 3.1 holds exactly when $P=P^{\star }$. All three assertions follow.

Gluing of ${\bar{\mathrm{\partial }}}_E$. First let ${\theta }_{\alpha },{\theta }_{\beta }$ be holomorphic frames of $E$ with transition matrix $T$, a holomorphic function with values in ${\mathrm{G}\mathrm{L}}_r(\mathbb{C})$ (Complex manifolds, holomorphic vector bundles and Hermitian metrics). For a local section $u$, the Leibniz rule for $\bar{\mathrm{\partial }}$ ([F1]) and $\bar{\mathrm{\partial }}T=0$ give $\bar{\mathrm{\partial }}(Tu)=\bar{\mathrm{\partial }}T\wedge u+T\bar{\mathrm{\partial }}u=T\bar{\mathrm{\partial }}u$, so the local $(p,q+1)$-forms computed in two frames agree. Next let $\phi$ be a holomorphic transition between two holomorphic charts and write $g=f\circ \phi$ for a coefficient function. Because $\phi$ is holomorphic, ${\bar{\mathrm{\partial }}}_{z_k^{\mathrm{\prime }}}{\phi }_j=0$ and ${\bar{\mathrm{\partial }}}_{z_k^{\mathrm{\prime }}}{\bar{\phi }}_j=\stackrel{‾}{{\mathrm{\partial }}_{z_k^{\mathrm{\prime }}}{\phi }_j}$ ([F1], [F2]); the chain rule [F2] therefore gives ${\bar{\mathrm{\partial }}}_{z_k^{\mathrm{\prime }}}g={\sum }_j\stackrel{‾}{{\mathrm{\partial }}_{z_k^{\mathrm{\prime }}}{\phi }_j}({\mathrm{\partial }}_{{\bar{z}}_j}f)(\phi )$. Since ${\phi }^{\ast }(d{\bar{z}}_j)={\sum }_k\overline{\mathrm{\partial }{z}_j\mathrm{/}\mathrm{\partial }{z}_k^{\mathrm{\prime }}}d{\bar{z}}_k^{\mathrm{\prime }}$, multiplying by $d{\bar{z}}_k^{\mathrm{\prime }}$ and summing reproduces the pullback of ${\sum }_j({\mathrm{\partial }}_{{\bar{z}}_j}f)d{\bar{z}}_j$; the holomorphic factors $d{z}^I$ pull back to $d{z}^{\mathrm{\prime }I}$-combinations, so bidegrees are preserved and ${\phi }^{\ast }\bar{\mathrm{\partial }}=\bar{\mathrm{\partial }}{\phi }^{\ast }$. Hence the local operators agree on overlaps and glue to a global smooth first-order operator ${\bar{\mathrm{\partial }}}_E$ that is independent of the atlas and frames.

The integral pairing on compactly supported smooth bundle sections is sesquilinear and conjugate symmetric by the fibre pairing and linearity of density integration. It is positive definite: a nonzero section has positive squared norm on a coordinate neighbourhood, hence positive integral. Fibrewise Cauchy–Schwarz follows by expanding $\mathrm{\parallel }s-\lambda t{\mathrm{\parallel }}^2\ge 0$ and minimizing over $\lambda$; the same expansion for the integral pairing gives $\mathrm{\mid }⟨s,t{⟩}_{L^2}\mathrm{\mid }\le \mathrm{\parallel }s{\mathrm{\parallel }}_2\mathrm{\parallel }t{\mathrm{\parallel }}_2$. Its norm therefore admits the Banach completion [F8]. For Cauchy sequences $s_j,t_j$, the estimate $\mathrm{\mid }⟨s_j,t_j⟩-⟨s_l,t_l⟩\mathrm{\mid }\le \mathrm{\parallel }{s}_j-s_l{\mathrm{\parallel }}_2\mathrm{\parallel }{t}_j{\mathrm{\parallel }}_2+\mathrm{\parallel }{s}_l{\mathrm{\parallel }}_2\mathrm{\parallel }{t}_j-t_l{\mathrm{\parallel }}_2$ makes their scalar pairings Cauchy and their limit independent of equivalent sequences. This limit is sesquilinear and conjugate symmetric, and its diagonal is ${\lim }_j\mathrm{\parallel }{s}_j{\mathrm{\parallel }}_2^2=\mathrm{\parallel }[s_j]{\mathrm{\parallel }}^2$. Thus it is a positive-definite inner product inducing the complete norm, so the completion is Hilbert. This also covers the zero bundle.

Vanishing of the square and bidegree action. In each holomorphic chart ${\bar{\mathrm{\partial }}}^2=0$ coefficientwise by [F1], and the gluing of step 1.1 identifies these local squares with ${\bar{\mathrm{\partial }}}_E^2$; hence ${\bar{\mathrm{\partial }}}_E^2=0$ globally. By construction ${\bar{\mathrm{\partial }}}_E$ raises the second bidegree by one and preserves the first; when $p>m$ or $q>m$ the bundles and hence both sides vanish by the degree convention.

Formal adjoint and its bidegree. The operator ${\bar{\mathrm{\partial }}}_E$ is a smooth first-order linear differential operator between the finite-rank Hermitian bundles ${\mathrm{\Lambda }}^{p,q}{T}^{\ast }X\otimes E$ and ${\mathrm{\Lambda }}^{p,q+1}{T}^{\ast }X\otimes E$ over $X$, whose density ${\mu }_g$ is smooth and positive, so [F5] supplies a unique smooth first-order formal adjoint ${\bar{\mathrm{\partial }}}_E^{\ast }$, characterized by the displayed pairing identity. It lowers the second bidegree by one: for a compactly supported smooth form $t$ of bidegree $(p,q+1)$ and $s$ of bidegree $(p^{\mathrm{\prime }},q^{\mathrm{\prime }})$, the pairing $⟨{\bar{\mathrm{\partial }}}_{E}s,t{⟩}_{L^2}$ is zero unless $(p^{\mathrm{\prime }},q^{\mathrm{\prime }}+1)=(p,q+1)$ by step 2.1 and orthogonality of distinct bidegrees, so ${\bar{\mathrm{\partial }}}_E^{\ast }t$ has no component in any bidegree other than $(p,q)$.

Formal symmetry and energy. Let $s,t$ be compactly supported smooth $E$-valued forms. Using the defining adjoint identities of step 3.1, $⟨{\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}s,t{⟩}_{L^2}=⟨{\bar{\mathrm{\partial }}}_E{\bar{\mathrm{\partial }}}_E^{\ast }s,t{⟩}_{L^2}+⟨{\bar{\mathrm{\partial }}}_E^{\ast }{\bar{\mathrm{\partial }}}_{E}s,t{⟩}_{L^2}=⟨{\bar{\mathrm{\partial }}}_E^{\ast }s,{\bar{\mathrm{\partial }}}_E^{\ast }t{⟩}_{L^2}+⟨{\bar{\mathrm{\partial }}}_{E}s,{\bar{\mathrm{\partial }}}_{E}t{⟩}_{L^2}\mathrm{.}$ Applying the pairing property of [F4] to each summand shows the right side is conjugate-symmetric in $s,t$, so $⟨{\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}s,t{⟩}_{L^2}=\stackrel{‾}{⟨{\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}t,s{⟩}_{L^2}}$; taking $s=t$ gives $⟨{\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}s,s{⟩}_{L^2}=\mathrm{\parallel }{\bar{\mathrm{\partial }}}_{E}s{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{\bar{\mathrm{\partial }}}_E^{\ast }s{\mathrm{\parallel }}_2^2\ge 0$. Thus ${\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}$ is formally symmetric and nonnegative in the sense of [F6], and by step 3.1 it preserves every bidegree $(p,q)$.

Principal symbol and Laplace type. Fix a point and compute in a chart and a unitary basis ([F7]); write ${\xi }^{0,1}$ for the $(0,1)$-part of a real covector $\xi$. In real coordinates the coefficient vectors of ${\bar{\mathrm{\partial }}}_E$ multiply to $\epsilon ({\xi }^{0,1})$, and those of ${\bar{\mathrm{\partial }}}_E^{\ast }$ to $-\epsilon ({\xi }^{0,1}{)}^{\ast }$, where $\epsilon$ is exterior multiplication and $\ast$ the metric adjoint; the metric adjoint of $\epsilon (\eta )$ deletes each factor of a wedge monomial pairing with $\eta$, with alternating signs, so the deletion check on unitary wedge monomials gives $\epsilon (\eta )\epsilon (\eta {)}^{\ast }+\epsilon (\eta {)}^{\ast }\epsilon (\eta )=\mathrm{\mid }\eta {\mathrm{\mid }}^2\mathrm{i}\mathrm{d}$, including the holomorphic $p$-factors and the bundle factor. Therefore the second-order coefficient of ${\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}$ is the scalar $\mathrm{\mid }{\xi }^{0,1}{\mathrm{\mid }}_g^2$, that is, $-\sum a^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_j$ with $a^{ij}{\xi }_i{\xi }_j=\mathrm{\mid }{\xi }^{0,1}{\mathrm{\mid }}_g^2$; this form is smooth in $x$ and positive definite on nonzero real $\xi$, since ${\xi }^{0,1}=0$ forces the conjugate $(1,0)$-part ${\xi }^{1,0}=0$ and hence $\xi =0$. By [F6] the operator ${\mathrm{\Delta }}_{{\bar{\mathrm{\partial }}}_E}$ is therefore Laplace type with the displayed principal symbol, and all assertions of the Definition are established.

Choose finitely many relatively compact bundle charts and subordinate smooth functions ${\rho }_i$ with sum one and supports strictly inside the charts. Each coefficient transformation on a compact overlap is multiplication by a smooth matrix followed by a smooth coordinate diffeomorphism with bounded derivatives and inverse derivatives at every fixed order. For a smooth coefficient function, the repeated chain rule expresses each derivative of the transformed function of order at most $k$ as a finite sum of bounded coefficients times derivatives of the original function of order at most $k$; change of variables bounds the integrals because the Jacobian and its inverse are bounded there. Approximate compactly supported Sobolev coefficients by [F1] smooth functions and pass these bounds and the weak derivative identities to the limit. Thus the same bounds hold on Sobolev classes. Multiplication by transition matrices is bounded by [F1]. Insert $u={\sum }_i{\rho }_{i}u$ to compare any two finite-chart norms; the finite sum yields bounds in both directions. The metric and density are bounded above and below on these compact supports, giving $H^0=L^2$ and equivalent norms.

Completeness is checked on the finitely many coefficient functions and derivatives: a Cauchy sequence has $L^2$ limits for every derivative of order at most $k$; passing the test-function identities to the limit identifies these as derivatives of the zeroth-order limit. The transition identities pass in $L^2$ by the bounds of step 1.1, so these local limits are one global section. This proves completeness for the sum-of-integral-derivatives inner product. To obtain smooth density, use cutoffs equal to one near each $\mathrm{supp}{\rho }_i$, mollify the zero-extended coefficient functions, multiply again by these cutoffs, extend the resulting sections by zero and sum; [F1] makes the sum converge in $H^k$. For a bounded $H^{k+1}$ sequence, all derivatives of the finitely many localized coefficients of order at most $k$ are bounded in $H^1$ on a fixed Euclidean ball. Apply [F3] successively to these finitely many families and retain a common subsequence converging in $L^2$ for every derivative. It is Cauchy in $H^k$, hence convergent. Apply the embedding of [F3] on the same balls and step 1.1 to get the $C^r$ bound when $k>r+n\mathrm{/}2$.

In a precompact chart each component of $Pu=f$ can be written $-{\mathrm{\partial }}_i(a^{ij}{\mathrm{\partial }}_j{u}_{\alpha })=f_{\alpha }-{\tilde{B}}_{\alpha \beta }^j{\mathrm{\partial }}_j{u}_{\beta }-C_{\alpha \beta }{u}_{\beta }\mathrm{.}$ The adjusted matrices $\tilde{B}$ include the derivatives of $a$. If $u\in H^1$, the right side is locally $L^2$ and bounded by $C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$ on an outer patch. The weak scalar equation follows by testing the original distributional equation against a compactly supported test in the single component and integrating its principal part once. [F2] therefore gives all components in $H^2$ on an inner patch, with the same bound. Cover $M$ by finitely many inner patches, each with a larger precompact outer patch, to get $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})\mathrm{.}$ The lower matrices have been moved to the source; no coupled principal system or Dirichlet theorem is imported.

After step 2.2, interpolation [F2] on the finitely many zero-extended localized coefficients gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le \epsilon \mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}+C_{\epsilon }\mathrm{\parallel }u{\mathrm{\parallel }}_2$; cutoff derivatives add only lower-order terms, absorbed by choosing their interpolation tolerances small. Choose $C\epsilon <1\mathrm{/}2$ to obtain the stated $k=0$ estimate. For induction, suppose $f\in H^k$ and $u\in H^{k+1}$, $k\ge 1$. On nested patches differentiate the scalar principal equation $k$ times. For $\mathrm{\mid }\alpha \mathrm{\mid }=k$, $w={\mathrm{\partial }}^{\alpha }{u}_{\beta }\in H^1$ satisfies the same scalar second-order equation with source ${\mathrm{\partial }}^{\alpha }{f}_{\beta }$ plus terms involving at most $k+1$ derivatives of $u$: in the commutator with the principal term at least one derivative hits $a$, and the lower terms have order at most one. All these terms are $L^2$, with bound $C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}})$. Apply [F2] to $w$ on successively smaller patches and sum to get $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}})$. Repeated use of the interpolation inequality of [F2] gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+1}}\le \epsilon \mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}+C_{k,\epsilon }\mathrm{\parallel }u{\mathrm{\parallel }}_2$; absorb again. Starting with step 2.2 and using $f\in H^k\subset H^{k-1}$ proves the required membership before this absorption. The embedding of step 2.1 then proves smoothness for smooth $f$.

Let $a(u,v)$ be the form obtained from $(Pu,v{)}_{L^2}$ by integrating the second-order principal term once, initially for smooth sections. It is bounded on $H^1\times H^1$: its integrands have at most one derivative in either factor and bounded coefficients. Formal symmetry and smooth density give $a(u,v)=\overline{a(v,u)}$. To prove a global lower bound, choose real smooth cutoffs ${\chi }_i$ supported in the charts with ${\sum }_i{\chi }_i^2=1$ (normalize a finite family of bumps by the positive smooth square root of their sum of squares). For smooth $u$, the principal integral of $a({\chi }_{i}u,{\chi }_{i}u)$ is bounded below by $c_i\mathrm{\parallel }\mathrm{\partial }({\chi }_{i}u){\mathrm{\parallel }}_2^2$: positive definiteness of the scalar coefficient and the positive bundle metric give this bound uniformly on its support. Every remaining term is bounded by $C\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_2\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_{H^1}+C\mathrm{\parallel }{\chi }_{i}u{\mathrm{\parallel }}_2^2$ and Young's inequality absorbs half the leading gradient. In the sum over $i$, $a(u,u)-{\sum }_{i}a({\chi }_{i}u,{\chi }_{i}u)$ contains only terms with a derivative on a cutoff; second derivatives on $u$ cancel by ${\sum }_i{\chi }_i^2=1$, and these remaining terms are bounded by $C\mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}+C\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. Absorb once more. Norm equivalence yields constants $c>0,C$ with $a(u,u)\ge c\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2-C\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{.}$ Density extends this real inequality and the bounded form to all $H^1$.

Fix $\lambda >C+1$. Then $a_{\lambda }=a+\lambda (\cdot ,\cdot {)}_2$ is coercive on $H^1$, so [F4] gives for every $f\in L^2$ a unique $u=Bf\in H^1$ with $a_{\lambda }(u,v)=(f,v{)}_2$ for every $v\in H^1$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le C^{\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Testing smoothly gives $Pu=f-\lambda u$ distributionally; step 3.1 gives $u\in H^2$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}\le C^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Conversely any $u\in H^2$ has $Pu\in L^2$ and integration by parts, passed by smooth approximation, gives the same weak identity with $f=(P+\lambda )u$. Uniqueness shows both inverse identities and $\mathrm{ran}B=H^2$. Symmetry gives $(Bf,g{)}_2=a_{\lambda }(Bf,Bg)=(f,Bg{)}_2$, so $B$ is bounded selfadjoint. Its range contains all smooth sections and is therefore dense; $Bf=0$ implies $(f,v{)}_2=0$ for every smooth $v$, hence $f=0$. Compactness follows from its bounded $H^2$ range and step 2.1.