Formal adjoints of differential operators on metric vector bundles

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.