Ordinary symbols, composition and proper localization

This bounded modified selection retains the arguments of AN03-U010, Symbols, operators and Sobolev scales, Sections 3–5, and AN03-U012, Detecting regularity without choosing coordinates, Sections 1–2 and 5. It specializes them to the ordinary class $S^m_{1,0}$ used in the frequency-graph lesson. The metric verification and the identification of the multiplier with the actual operator integral are supplied explicitly. The general manifold and coordinate-transport theorems remain separate.

Original principal author and publisher: AN-03 course-writing task / AN-03 local course project, 2026. Earlier modification: AN-03 course-writing task and OpenAI Codex. This selection and its connecting arguments: GPT-6 Astra (OpenAI), Ultra, 4 October 2026; publisher: AN-04 local course project.

Original text: CC0.

O0. Exact prerequisites and conventions

The quadratic multiplier companion includes its complete localization and finite-dimensional prerequisites. The measure and Fourier companions supply completed integration, inversion and distributional compatibility. Use $D=-i\partial$ , Fourier exponential $e^{-ix\cdot\xi}$ , and inverse factor $(2\pi)^{-n}$ . Matrix symbols have fixed finite sizes and are multiplied in the displayed order.

Write $a\in S^m$ when every derivative satisfies

|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)| \leq C_{\alpha,\beta}\langle\xi\rangle^{m-|\alpha|}. \tag{O1}

These are global estimates in this paragraph. On an open base set the same bounds are required on each compact subset. A compact base cutoff then extends a local symbol by zero to a global symbol. Orders are real. All statements include dimension zero, where the integrals have one point, Fourier maps are the identity and symbols are finite matrices.

O0.1. Product tests determine a distributional kernel

Let $\varphi(x,y)$ be compact smooth in a product of two Euclidean open sets. Choose compact smooth $\alpha(x),\beta(y)$ , each one near the corresponding projection of its support. Extend $\varphi$ by zero to Euclidean space. Fourier inversion gives

\varphi(x,y)=(2\pi)^{-2n}\iint \widehat\varphi(\xi,\eta) \alpha(x)e^{ix\cdot\xi}\, \beta(y)e^{iy\cdot\eta}\,d\xi\,d\eta .

The same integral converges uniformly after every fixed number of base derivatives: its Fourier transform is Schwartz, so all resulting polynomial factors are integrable. Truncate to larger frequency boxes and approximate each compact integral by Riemann sums. Uniform continuity on the compact base and frequency product makes these sums converge with any fixed finite set of derivatives. At step $k$ make the error less than $1/k$ for all derivatives through order $k$ . The resulting finite sums of product test functions have one fixed compact support and converge to $\varphi$ in its complete smooth test topology. Thus a distribution vanishing on all product tests vanishes on $\varphi$ . This proves the kernel detection used in O5–O6 without a general kernel representation theorem.

O1. Schwartz action and the exact multiplier formula

For $u\in\mathcal S$ define

\operatorname{Op}(a)u(x)=(2\pi)^{-n} \int e^{ix\cdot\xi}a(x,\xi)\widehat u(\xi)\,d\xi . \tag{O2}

This integral and all its differentiated versions converge absolutely. Output derivatives create finitely many powers of $\xi$ and base derivatives of $a$ . To estimate $x^\gamma\partial_x^\beta\operatorname{Op}(a)u$ , move $|\gamma|$ frequency derivatives off the exponential by integration by parts. Each resulting integrand is a symbol derivative, a polynomial and a derivative of $\widehat u$ . Its integral is bounded by one sufficiently large Schwartz seminorm of $u$ times finitely many symbol seminorms, with a remaining integrable weight of order less than $-n$ . Boundary terms vanish by the same rapid decrease. Thus (O2) maps $\mathcal S$ continuously to itself, with uniform operator seminorms on each bounded symbol set. Differentiating and integrating once also gives

[\operatorname{Op}(a),D_j]=i\operatorname{Op}(\partial_{x_j}a), \qquad [\operatorname{Op}(a),x_j]=-i\operatorname{Op}(\partial_{\xi_j}a). \tag{O3}

For $c\in\mathcal S(\mathbb R^{2n})$ , let $Qc$ be the Fourier multiplier $e^{ip\cdot q}$ in the variables dual to $(y,\eta)$ . The partial Fourier transform in $y$ is a Schwartz function of $(p,\eta)$ : multiplication by $p$ is integration by parts in $y$ , differentiation in $p$ inserts powers of $y$ , and every seminorm is bounded by integrating an arbitrarily decreasing $y$ weight. The same holds for its inverse. Their two-sided identities follow from the supplied Fourier inversion theorem on each slice; the seminorm estimates justify every parameter derivative. The Fourier translation rule in $\eta$ therefore gives

\begin{split} Qc(y,\eta) &=(2\pi)^{-n}\int e^{iy\cdot p} \mathcal F_yc(p,\eta+p)\,dp\\ &=(2\pi)^{-n}\iint e^{-iv\cdot w} c(y+v,\eta+w)\,dv\,dw . \end{split} \tag{O4}

For the second line expand the partial transform and substitute $p=w$ , its original base variable $y+v$ . That double integral is absolutely convergent for a Schwartz $c$ . This establishes the multiplier normalization and sign without a kernel theorem.

O2. The exact ordinary metric verification

In $(y,\eta)$ put

g_{(y,\eta)}(v,\nu)=|v|^2+\langle\eta\rangle^{-2}|\nu|^2, \qquad m(y,\eta)=\langle\eta\rangle^r . \tag{O5}

The function $\langle\eta\rangle$ is 1-Lipschitz: the reverse triangle inequality applied to $(1,\eta)$ proves it. Thus, within a metric ball of radius less than $1/2$ , the neighboring frequency weights have ratio between $1/2$ and $3/2$ . This proves slow variation and local weight comparison, uniformly in the center.

For $A(p,q)=p\cdot q$ , the symmetric map is $\frac12\left(\begin{smallmatrix}0&I\\I&0\end{smallmatrix}\right)$ . The companion's dual-form calculation gives

g_Y^A(X-Y)=4\bigl(\langle\eta_Y\rangle^2|y_X-y_Y|^2 +|\eta_X-\eta_Y|^2\bigr), \qquad h(X)=\frac1{2\langle\eta_X\rangle}\leq\frac12 . \tag{O6}

In particular $g_X\leq g_X^A$ . For $u=\eta_X,v=\eta_Y$ , $\langle u\rangle\leq\langle v\rangle+|u-v| \leq\langle v\rangle(1+|u-v|)$ . Interchange $u,v$ for the reciprocal ratio. Squaring and using $(1+t)^2\leq2(1+t^2)$ proves

g_Y(T)\leq2(1+|\eta_X-\eta_Y|^2)g_X(T),\qquad \frac{m(Y)}{m(X)} \leq2^{|r|/2}(1+|\eta_X-\eta_Y|^2)^{|r|/2}. \tag{O7}

These are precisely the observation-point assumptions of the quadratic-multiplier Theorems 7.1 and 8.1, with constants independent of $X,Y$ . Consequently

Qc-\sum_{|\alpha|<N} \frac{\partial_\eta^\alpha D_y^\alpha c}{\alpha!} \in S^{r-N},\qquad c\in S^r. \tag{O8}

The coefficient follows by expanding $(iD_y\cdot D_\eta)^j/j!$ ; since $iD_\eta=\partial_\eta$ , it has exactly the displayed sign. Every derivative has its own finite input-seminorm bound. Convergence on bounded symbol sets in the local smooth topology is preserved, as proved in the companion.

For parameters $(x,\xi)$ , apply this to $c_{x,\xi}(y,\eta)=a(x,\eta)b(y,\xi)$ . Its weight is $\langle\eta\rangle^{m_1}\langle\xi\rangle^{m_2}$ . A parameter derivative $\partial_\xi^\alpha\partial_x^\beta$ lowers the latter order by $|\alpha|$ ; a transformed derivative $\partial_\eta^{\alpha'}\partial_y^{\beta'}$ lowers the former by $|\alpha'|$ . Thus the full pre-diagonal remainder is bounded by

C\langle\eta\rangle^{m_1-N-|\alpha'|} \langle\xi\rangle^{m_2-|\alpha|}. \tag{O9}

Parameter derivatives are genuine derivatives of the resulting function: apply the compact symbol approximants from the metric companion, use the estimates for every parameter derivative, and pass to the limit in the fundamental theorem along compact parameter segments. One additional derivative bounds the modulus of continuity; a finite grid upgrades pointwise convergence to uniform convergence on each compact set. Repeating this proves the assertion in every order. All constants use finitely many original seminorms.

O3. Adjoints, composition and tempered distributions

For Schwartz symbols, Fubini applied to (O2) twice gives the product symbol

(a\circ b)(x,\xi)=(2\pi)^{-n} \iint e^{i(x-y)\cdot(\eta-\xi)} a(x,\eta)b(y,\xi)\,dy\,d\eta =\left[Q_{y,\eta}c_{x,\xi}\right]_{y=x,\eta=\xi}. \tag{O10}

For example $a=a(\xi)$ , $b=e^{iv\cdot x}b_0(\xi)$ gives $a\circ b=e^{iv\cdot x}a(\xi+v)b_0(\xi)$ , confirming the sign. For an adjoint, interchanging the two base variables and conjugating the scalar pairing gives, by the same integral (O4),

a^\dagger=Q(a^*). \tag{O11}

Here $^*$ is conjugate transpose and the pairing is $(u,v)=\int u\overline v$ , linear in $u$ . Explicitly, the adjoint kernel has integrand $e^{i(x-y)\cdot\xi}a(y,\xi)^*$ ; its left symbol is obtained by putting $y=x+v,\xi=\eta+w$ , giving $e^{-iv\cdot w}$ in (O4).

Equations (O8)–(O9), followed by the diagonal chain rule, prove

\begin{aligned} a^\dagger&\in S^{m_1},\qquad a\circ b\in S^{m_1+m_2},\\ a^\dagger-\sum_{|\alpha|<N} \frac{\partial_\xi^\alpha D_x^\alpha(a^*)}{\alpha!} &\in S^{m_1-N},\\ a\circ b-\sum_{|\alpha|<N} \frac{(\partial_\xi^\alpha a)(D_x^\alpha b)}{\alpha!} &\in S^{m_1+m_2-N}. \end{aligned} \tag{O12}

For a total frequency derivative of the diagonal product, each derivative lands on either $\xi$ or $\eta$ in (O9); either choice lowers the combined order by one. Base derivatives have order zero.

These formulas and the operator identities hold for all ordinary symbols. To justify this passage, multiply symbols by cutoffs $\chi(\epsilon x)\chi(\epsilon\xi)$ , with $\chi=1$ near zero. These are bounded approximants in the original symbol classes: on the support of a differentiated frequency cutoff, $\epsilon\leq C\langle\xi\rangle^{-1}$ , and all base cutoff derivatives are bounded. They converge locally smoothly. The multiplier theorem gives bounded, locally convergent adjoints and products in their asserted orders. For a fixed Schwartz $u$ , (O2) and dominated convergence give local smooth convergence of the outputs. O1 bounds every stronger output Schwartz seminorm uniformly; outside a large base ball one extra decay power makes the tails small. Hence the outputs converge in every Schwartz seminorm. The same uniform operator bounds permit passage through the double composition. The adjoint pairing and

\operatorname{Op}(a)\operatorname{Op}(b) =\operatorname{Op}(a\circ b) \tag{O13}

therefore hold on $\mathcal S$ .

Define the extension to the usual bilinear distribution dual by $\langle Pu,\phi\rangle=\langle u,\overline{P^\dagger\overline\phi}\rangle$ . O1 and (O11) make the right test function Schwartz and give continuity. On ordinary functions this is exactly the original integral pairing; thus no Fourier or duality convention has changed. Transposing (O13) proves it on $\mathcal S'$ . The same proof supplies finite-seminorm continuity of the adjoint and bilinear product maps. No general bijection between all tempered kernels and all continuous operators is required for these ordinary symbols.

O4. Compact amplitudes and smooth off-diagonal kernels

For a smooth $c(x,y,\eta)$ , compactly supported in $(x,y)$ , with ordinary order $m$ in $\eta$ , the distribution

K_c(x,y)=(2\pi)^{-n}\int e^{i(x-y)\cdot\eta}c(x,y,\eta)\,d\eta \tag{O14}

is well defined. Against a compact smooth test function, integrate by parts in $y$ with $(1-\Delta_y)^k$ ; division by $\langle\eta\rangle^{2k}$ , $2k>m+n$ , makes the integral absolutely convergent and controlled by finitely many test seminorms. Derivatives of $c$ in $y$ have the same order. A frequency cutoff can therefore be removed by dominated convergence.

Apply O2 in $(y,\eta)$ , with $x$ as parameter, and restrict:

p(x,\xi)=\left[Q_{y,\eta}c(x,y,\eta)\right]_{y=x,\eta=\xi}, \qquad p-\sum_{|\alpha|<N} \frac{\partial_\eta^\alpha D_y^\alpha c(x,x,\xi)}{\alpha!} \in S^{m-N}. \tag{O15}

For compact frequency support, Fourier inversion in $x-y$ identifies the left symbol with (O4), so its operator kernel is (O14). For general amplitudes use frequency cutoffs as in O3: the amplitudes are bounded in their original estimates, (O15) converges locally with all derivatives and stays bounded in $S^m$ , and testing the kernels in the absolutely convergent regularization above gives the same distributional limit. This proves both kernel identity and every remainder estimate.

For a left symbol $p$ , its kernel is smooth off $x=y$ . On a compact set with $|x-y|\geq\delta>0$ , repeatedly use

e^{i(x-y)\cdot\xi} =|x-y|^{-2k}(-\Delta_\xi)^k e^{i(x-y)\cdot\xi}. \tag{O16}

Moving the derivatives onto the symbol lowers its order by $2k$ . After any prescribed number of $x,y$ derivatives, choose $k$ large enough to make the integral absolutely convergent. Coefficients and their derivatives are uniformly bounded on the separated compact set, so dominated convergence proves all derivatives there. A symbol in every negative order has a smooth kernel on every compact set, by the same argument without division by $|x-y|$ .

O5. Proper operators and the actual local representation

Work in a Euclidean open set. An ordinary pseudodifferential kernel is smooth off the diagonal and, near each diagonal base point, has a local symbol representation (O14), of order $m$ , modulo a smooth kernel. A compact restriction of the base variables is understood before applying O14. This is the local class used here. It includes each quantized symbol, by O4. Multiplication by smooth base cutoffs and O15 preserve the class and its order.

A kernel is proper when the inverse image of a compact set under either coordinate projection of its support is compact. For a compact output set this confines all relevant input points to a compact set; reversing the projections gives the analogous output bound for a compact input set. Choose cutoffs equal to one on neighborhoods of those compact sets.

For each fixed output cutoff $\chi$ , take such an input cutoff $\psi$ , and a larger $\psi_1=1$ near $\operatorname{supp}\psi$ . The compact kernel $\chi(x)K(x,y)\psi_1(y)$ has a finite cover near its diagonal part by local representations. A finite smooth partition in the base exists by the supplied U001 cutoff construction. Multiply by a cutoff in the second variable equal to one near each diagonal base piece. Each resulting local amplitude has compact support and reduces by O15 to a compact-base symbol of order $m$ . All omitted pieces are smooth by O16 and have compact support in both variables. Summing the finite symbol pieces gives $p\in S^m$ and a compact smooth kernel $R$ with

\chi Pu=\operatorname{Op}(p)(\psi u)+R(\psi u). \tag{O17}

Indeed $\psi_1\psi=\psi$ , and proper support makes every discarded input term vanish near $\operatorname{supp}\chi$ . This is the representation assumed in the endpoint companion B6.

Here is the distributional meaning of that assertion. The transpose of each local symbol operator sends compact smooth tests to smooth functions, by O1–O3 after localization. The smooth remainder has the same property by differentiation under its integral. Proper support confines the support of the transposed output to one compact set, for tests supported in a fixed compact set. Its derivative seminorms are bounded by finitely many seminorms of the input test function. Thus it is a continuous map of the compact smooth test spaces. The usual inductive-limit test topology consequently makes the transpose-test map continuous on $\mathcal D$ . Its transpose defines $P:\mathcal D'\to\mathcal D'$ . This also gives strong-dual continuity: for a bounded test set $B$ , the seminorm of $Pu$ on $B$ is the seminorm of $u$ on its transpose image; a continuous linear map takes bounded sets to bounded sets, directly by pulling back each zero neighborhood in the definition of boundedness. If $\psi=1$ near the corresponding input compact, pairing with each output test shows $Pu=P(\psi u)$ there. This proves the cutoff independence and (O17) on distributions. The same argument gives $P:C^\infty\to C^\infty$ ; the reverse proper projection keeps outputs compact for compactly supported inputs. It also proves $\mathcal E'\to\mathcal E'$ , where $\mathcal E'$ denotes compactly supported distributions.

O6. Proper products, remainders and the principal symbol

Properness makes composition meaningful on all distributions. For compact output and input cutoffs $\phi,\psi$ , choose an intermediate cutoff $\chi$ equal to one near both supports. Then

\phi AB\psi=(\phi A\chi)(\chi B\psi) +\phi A(1-\chi^2)B\psi. \tag{O18}

The first term uses the finite local representations of O5 and the product formula O12. For the second, properness confines the intermediate variable to a compact set. Where $1-\chi^2$ is nonzero, both relevant external variables are separated from it; O16 makes both kernels smooth there. Their compact integral and every derivative are therefore smooth. Smooth remainders in the first term also remain smooth: one operator acts on a compact smooth kernel in one variable, and its O1 bounds, with any finite parameter derivatives, justify differentiation. Transposition treats the other order. Smooth proper kernels consequently form a two-sided ideal.

For completeness, the support of $AB$ is contained in the composed support relation. To see absence of support at an external pair outside that relation, properness confines all possible intermediate points near that pair to a compact set. Each intermediate point misses one of the two closed supports; finite subcovering and a smooth partition split the pairing into terms in which one factor vanishes. This proves the assertion first on tests and then by transpose on distributions. The composed relation is closed: for convergent external pairs, properness confines the intermediate points to a compact set, so a subsequence converges into both closed supports. Both its projections are proper by the same two successive compactness bounds. Thus $AB$ is proper.

The $N=1$ formula in O12 gives principal symbol $ab$ modulo $S^{m+m'-1}$ , in the displayed matrix order. For scalars the leading products commute; the $N=2$ formula gives

\sigma_{m+m'-1}([A,B])=\frac1i \sum_j(\partial_{\xi_j}a\,\partial_{x_j}b -\partial_{x_j}a\,\partial_{\xi_j}b) \pmod{S^{m+m'-2}} . \tag{O19}

For matrices the order $m+m'$ term is $ab-ba$ . In particular, if an order-one matrix symbol vanishes on a specified frequency graph modulo $S^0$ , its commutator with an order-zero matrix symbol still has order at most one and vanishes there modulo $S^0$ . The derivative and lower-order terms are of order zero. No order-zero matrix-commutator assertion is made.

Free human comparison

Nicolas Lerner's author-hosted PSEUDO.2005.pdf, Section 1.2, especially (1.2.2) and (1.2.8), gives the quantization and composition formulas with its $2\pi$ Fourier convention. Its confinement and symbolic-calculus Sections 3.2–4.1 are the free comparison for the retained multiplier route. All used proofs and the conversion of conventions are supplied above and in the linked programme companions. No human-source expression is copied.