[proofplan] The proof is local because principal symbols are defined in coordinate charts and then patched by their coordinate transformation law. In one coordinate patch, write the density as $w(x)\,d\mathcal{L}^n(x)$ with $w$ positive and smooth. The adjoint with respect to this weighted pairing is $w^{-1}$ times the Euclidean formal adjoint followed by multiplication by $w$. The standard local adjoint symbol expansion shows that the leading homogeneous term is the complex conjugate of the leading homogeneous term of $A$, while every term involving a derivative in $\xi$ or a derivative of $w$ drops the order by at least one. [/proofplan]

[step:Localize the adjoint computation to one coordinate patch] Let $(U,\varphi)$ be a coordinate chart on $M$, with $\varphi(U) \subset \mathbb{R}^n$ open. Let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$, restricted to the Borel subsets of $\varphi(U)$ when used on the coordinate domain. After inserting compactly supported cutoffs in $U$, it is enough to prove the asserted principal-symbol identity for the localized operator in these coordinates, because the principal symbol of a scalar classical pseudodifferential operator is determined locally and transforms invariantly on $T^*M \setminus \{0\}$. In the chart, write the density as \begin{align*} d\mu(x)=w(x)\,d\mathcal{L}^n(x) \end{align*} where $w:\varphi(U)\to(0,\infty)$ is smooth. Let \begin{align*} a:\varphi(U)\times \mathbb{R}^n \to \mathbb{C} \end{align*} be a classical symbol of order $m$ for the localized operator $A$ in the standard scalar quantization, so that its homogeneous principal part is denoted by $a_m(x,\xi)$ for $\xi \ne 0$. Thus, in this coordinate patch, \begin{align*} \sigma_m(A)(x,\xi)=a_m(x,\xi). \end{align*} Proper support ensures that the localized kernels have compact support in the relevant variables after cutoffs are inserted, so the formal adjoint is computed by transposing and conjugating the local kernel without boundary terms or support problems. [/step]

[step:Express the weighted formal adjoint through the Euclidean formal adjoint]Let $M_w:C_c^\infty(\varphi(U))\to C_c^\infty(\varphi(U))$ denote multiplication by $w$, so $M_w f=w f$. Let $A^\dagger$ denote the formal adjoint of the localized operator $A$ with respect to the Euclidean pairing \begin{align*} (f,g)_{L^2(\varphi(U),\mathcal{L}^n)}=\int_{\varphi(U)} f(x)\,\overline{g(x)}\,d\mathcal{L}^n(x). \end{align*} Then the formal adjoint with respect to $w\,d\mathcal{L}^n$ is \begin{align*} A^*=M_{w^{-1}} A^\dagger M_w \end{align*} on compactly supported test functions in the chart. Indeed, for $u,v\in C_c^\infty(\varphi(U))$, the defining identity for $A^\dagger$ gives \begin{align*} \int_{\varphi(U)} (Au)(x)\,\overline{v(x)}\,w(x)\,d\mathcal{L}^n(x)=\int_{\varphi(U)} (Au)(x)\,\overline{(M_wv)(x)}\,d\mathcal{L}^n(x). \end{align*} Therefore \begin{align*} \int_{\varphi(U)} (Au)(x)\,\overline{(M_wv)(x)}\,d\mathcal{L}^n(x)=\int_{\varphi(U)} u(x)\,\overline{(A^\dagger M_wv)(x)}\,d\mathcal{L}^n(x). \end{align*} Since $w(x)>0$, this last integral equals \begin{align*} \int_{\varphi(U)} u(x)\,\overline{(M_{w^{-1}}A^\dagger M_wv)(x)}\,w(x)\,d\mathcal{L}^n(x). \end{align*} This is exactly the defining relation for the weighted formal adjoint.[/step]

[guided]The weighted pairing differs from the Euclidean pairing only by the positive smooth factor $w$. The goal is to move that factor into the operator and then use the ordinary Euclidean adjoint. Define $M_w:C_c^\infty(\varphi(U))\to C_c^\infty(\varphi(U))$ by $M_w f=w f$. Since $w$ is smooth and positive, multiplication by $w^{-1}$ is also a smooth multiplication operator. Let $A^\dagger$ be the formal adjoint of the localized operator $A$ with respect to the Euclidean measure $\mathcal{L}^n$, meaning \begin{align*} \int_{\varphi(U)} (Au)(x)\,\overline{g(x)}\,d\mathcal{L}^n(x)=\int_{\varphi(U)} u(x)\,\overline{(A^\dagger g)(x)}\,d\mathcal{L}^n(x) \end{align*} for all $u,g\in C_c^\infty(\varphi(U))$. Now take $g=M_wv$. Because $w$ is real-valued and positive, $\overline{(M_wv)(x)}=w(x)\overline{v(x)}$. Hence \begin{align*} \int_{\varphi(U)} (Au)(x)\,\overline{v(x)}\,w(x)\,d\mathcal{L}^n(x)=\int_{\varphi(U)} (Au)(x)\,\overline{(M_wv)(x)}\,d\mathcal{L}^n(x). \end{align*} Applying the Euclidean adjoint identity to the right-hand side gives \begin{align*} \int_{\varphi(U)} (Au)(x)\,\overline{(M_wv)(x)}\,d\mathcal{L}^n(x)=\int_{\varphi(U)} u(x)\,\overline{(A^\dagger M_wv)(x)}\,d\mathcal{L}^n(x). \end{align*} To return to the weighted measure, insert $w(x)w(x)^{-1}=1$ inside the last integrand: \begin{align*} \int_{\varphi(U)} u(x)\,\overline{(A^\dagger M_wv)(x)}\,d\mathcal{L}^n(x)=\int_{\varphi(U)} u(x)\,\overline{(M_{w^{-1}}A^\dagger M_wv)(x)}\,w(x)\,d\mathcal{L}^n(x). \end{align*} Thus the weighted formal adjoint is precisely $M_{w^{-1}}A^\dagger M_w$. This formula is useful because multiplication by a smooth function has order $0$, so it can affect lower-order symbol terms but cannot change the top-order adjoint mechanism except through multiplication by its leading symbol.[/guided]

[step:Compute the leading symbol of the Euclidean adjoint] We use the local adjoint symbol formula for scalar classical pseudodifferential operators in the chosen standard quantization. Its hypotheses apply because the localized operator has a classical full symbol $a \in S_{\mathrm{cl}}^m(\varphi(U) \times \mathbb{R}^n)$ and is properly supported after the cutoffs inserted in the first step. Therefore the Euclidean formal adjoint $A^\dagger$ is a classical pseudodifferential operator of order $m$, and its full local symbol $a^\dagger(x,\xi)$ has an asymptotic expansion of the form \begin{align*} a^\dagger(x,\xi)\sim \sum_{\alpha\in\mathbb{N}_0^n}\frac{1}{\alpha!}\,\partial_\xi^\alpha D_x^\alpha \overline{a(x,\xi)} \end{align*} up to the conventional harmless powers of $i$ determined by the chosen quantization. Here $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index and $D_x^\alpha$ denotes the corresponding normalized coordinate derivative in the symbolic calculus convention. The term with $\alpha=0$ is $\overline{a(x,\xi)}$. Its homogeneous component of degree $m$ is $\overline{a_m(x,\xi)}$. If $|\alpha|\geq 1$, then $\partial_\xi^\alpha$ lowers the symbol order by $|\alpha|$, while $D_x^\alpha$ does not increase the order. Hence every term with $|\alpha|\geq 1$ has order at most $m-1$. Therefore the principal local symbol of $A^\dagger$ is \begin{align*} \sigma_m(A^\dagger)(x,\xi)=\overline{a_m(x,\xi)}. \end{align*} [/step]

[step:Show that the density weight changes only lower-order terms] The local weighted adjoint is $M_{w^{-1}}A^\dagger M_w$. Multiplication by $w$ is a classical pseudodifferential operator of order $0$ with principal symbol $w(x)$, and multiplication by $w^{-1}$ is a classical pseudodifferential operator of order $0$ with principal symbol $w(x)^{-1}$. We apply the local composition formula for classical pseudodifferential operators to the three properly supported factors $M_{w^{-1}}$, $A^\dagger$, and $M_w$. The multiplication operators have order $0$, while $A^\dagger$ has order $m$, so the product has order $m$ and its principal symbol is the product of the principal symbols: \begin{align*} \sigma_m(M_{w^{-1}}A^\dagger M_w)(x,\xi)=w(x)^{-1}\,\sigma_m(A^\dagger)(x,\xi)\,w(x). \end{align*} Substituting the previous step gives \begin{align*} \sigma_m(M_{w^{-1}}A^\dagger M_w)(x,\xi)=w(x)^{-1}\,\overline{a_m(x,\xi)}\,w(x)=\overline{a_m(x,\xi)}. \end{align*} All other terms in the composition formula contain at least one derivative in $\xi$ of one of the symbols or at least one derivative of the smooth weight. Such terms have order at most $m-1$, so they do not contribute to the principal symbol. [/step]

[step:Patch the local identity to obtain the global principal symbol] The preceding computation holds in every coordinate patch and for every compactly supported localization. Since $A$ is properly supported, these local adjoint computations patch to the global formal adjoint on $C_c^\infty(M)$ without support obstructions. The local symbolic calculus also shows that the patched operator $A^*$ is classical of order $m$. In each chart, the principal symbol of $A^*$ is the complex conjugate of the local principal symbol of $A$. The scalar principal symbol is invariant under coordinate changes as a homogeneous function on $T^*M\setminus\{0\}$, and complex conjugation commutes with the transition maps because those transition maps act by pullback on the cotangent variable and do not alter scalar complex conjugation. Hence the local identities assemble to the global identity \begin{align*} \sigma_m(A^*)(x,\xi)=\overline{\sigma_m(A)(x,\xi)} \end{align*} for every $(x,\xi)\in T^*M\setminus\{0\}$. This proves both $A^*\in\Psi_{\mathrm{cl}}^m(M)$ and the asserted formula for its principal symbol. [/step]