Wavefront sets, Hadamard kernels and admissible local products

In a chart, localize $u$ by a cutoff $\chi$ and write $\widehat{\chi u}(\xi )=⟨u,\chi (x)e^{-2\pi ix\cdot \xi }\mathrm{\mid }dx\mathrm{\mid }⟩$. The finite-order bound [F3] gives polynomial growth in $\xi$ by differentiating the test at most its fixed order; parameter derivatives multiply it by powers of $x$, which remain bounded on the fixed compact support. Difference quotients converge in each fixed-support test seminorm by the Taylor remainder bound, so pairing shows that $\widehat{\chi u}$ is smooth. This is also the localized Fourier transform in [F4].

Suppose $(p,{\xi }_0)$ is regular for $u$ in the $x$ chart, witnessed by a cutoff ${\chi }_x$ and an open cone ${\mathrm{\Gamma }}_x$. Let $y=\psi (x)$ be a smooth coordinate change, restrict to a compact patch about $p$, and put ${\eta }_0=(D\psi (p){)}^{-T}{\xi }_0$. Shrink the patch and choose nested cones ${\mathrm{\Gamma }}_0⋐{\mathrm{\Gamma }}_1⋐{\mathrm{\Gamma }}_x$ around ${\xi }_0$ and a target cone $V$ around ${\eta }_0$ so that $D\psi (x{)}^{T}V\subset {\mathrm{\Gamma }}_0$ throughout the patch. Choose a target cutoff ${\chi }_y$ equal to one near $p$ and supported where ${\chi }_x=1$, using [F8]. The target localized Fourier value is the source distribution paired with $a_{\eta }(x)={\chi }_y(\psi (x))\mathrm{\mid }\det D\psi (x)\mathrm{\mid }{e}^{-2\pi i\psi (x)\cdot \eta }$; because ${\chi }_x=1$ on its support this is also the pairing with ${\chi }_{x}u$. Fourier inversion [F5], passed through this compactly supported distribution by [F6], expresses the value as $\int \widehat{{\chi }_{x}u}(\xi )\widehat{a_{\eta }}(-\xi )d\xi$. The phase in $\widehat{a_{\eta }}(-\xi )$ is $2\pi (x\cdot \xi -\psi (x)\cdot \eta )$, with gradient $2\pi (\xi -D\psi (x{)}^T\eta )$. In the near-stationary region $\mathrm{\mid }\xi -D\psi (x{)}^T\eta \mathrm{\mid }\le c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$, for sufficiently small $c>0$, cone separation gives $\xi \in {\mathrm{\Gamma }}_x$ and $\mathrm{\mid }\xi \mathrm{\mid }\asymp \mathrm{\mid }\eta \mathrm{\mid }$; there $\widehat{{\chi }_{x}u}(\xi )$ decreases faster than every power, while $\mathrm{\mid }\widehat{a_{\eta }}(-\xi )\mathrm{\mid }\le \mathrm{\parallel }{a}_{\eta }{\mathrm{\parallel }}_1$ is uniformly bounded, and the region has only polynomial volume in $\mathrm{\mid }\eta \mathrm{\mid }$. On the complementary region the phase gradient is bounded below by $c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$; repeated integration by parts gives arbitrary powers of joint decay for $\widehat{a_{\eta }}(-\xi )$. The polynomial bound [F3] for $\widehat{{\chi }_{x}u}$ then makes this part rapidly decreasing after integration. This proves regularity in the target chart.

Applying step 2.1 also to ${\psi }^{-1}$ proves equivalence of regularity in both charts, and [F7] gives exactly $\eta =(D\psi {)}^{-T}\xi$. Regularity is open because the same cutoff and a smaller cone work for nearby base points; positive rescaling of a covector does not change rapid decay. Hence its complement is closed and conic.

Suppose ${\chi }_1$ is a regularity witness for $u$ on an open cone $\mathrm{\Gamma }$, and let $\rho \in C_c^{\mathrm{\infty }}$ have support where ${\chi }_1=1$. If ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ is an open cone whose closure of directions on the unit sphere is contained in $\mathrm{\Gamma }$, then $\widehat{\rho u}$ is rapidly decreasing on ${\mathrm{\Gamma }}^{\mathrm{\prime }}$. Indeed, $\rho u=\rho ({\chi }_{1}u)$ and Fourier inversion gives $\widehat{\rho u}=\widehat{\rho }\ast \widehat{{\chi }_{1}u}$; passing the Schwartz integral through the compactly supported distribution uses [F6]. Repeated integration by parts makes $\widehat{\rho }$ rapidly decreasing, and [F3] bounds $\widehat{{\chi }_{1}u}$ polynomially. Choose an intermediate cone with closure of directions between those of ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ and $\mathrm{\Gamma }$. In the convolution, for frequencies $\eta$ in this intermediate cone with $\mathrm{\mid }\eta \mathrm{\mid }\ge \mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, the second factor is rapidly decreasing at frequency comparable to $\mathrm{\mid }\xi \mathrm{\mid }$. If $\mathrm{\mid }\eta \mathrm{\mid }<\mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, then $\mathrm{\mid }\xi -\eta \mathrm{\mid }>\mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$; for the remaining frequencies, $\eta$ lies outside the intermediate cone and $\mathrm{\mid }\eta \mathrm{\mid }\ge \mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}2$, so angular separation gives $\mathrm{\mid }\xi -\eta \mathrm{\mid }\ge c(\mathrm{\mid }\xi \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$ for some $c>0$. In these latter regions the first factor is rapidly decreasing, while [F3] bounds the other factor polynomially. These estimates give the arbitrary-power bound for $\mathrm{\mid }\xi \mathrm{\mid }\ge 1$ on ${\mathrm{\Gamma }}^{\mathrm{\prime }}$, and [F4] gives boundedness on bounded sets. The convolution is absolutely integrable by the same estimates, and [F9] justifies its Fourier-integral identity. Applying this cutoff transfer to a common $\rho$ for witnesses for $u$ and $v$, and intersecting their smaller cones, gives rapid decay for $\widehat{\rho (u+v)}=\widehat{\rho u}+\widehat{\rho v}$. This proves $\mathrm{WF}(u+v)\subseteq \mathrm{WF}(u)\cup \mathrm{WF}(v)$. For a diffeomorphism, step 2.1 gives the cotangent image inclusion for pullback; applying the same argument to ${\psi }^{-1}$ gives the reverse inclusion.

If $u$ is smooth, integration by parts in a compact chart cutoff gives rapid decay in every cone. Conversely, suppose $\mathrm{WF}(u)=\mathrm{\varnothing }$ and fix a point $p$ in a chart. In dimension zero every distribution is locally a scalar function, so assume the chart dimension is at least one. Form the set of all open cones ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ for which there exist a regularity witness $(\chi ,\mathrm{\Gamma })$ at $p$ with the closure of the directions of ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ on the unit sphere contained in $\mathrm{\Gamma }$. This set covers the unit covector sphere: every direction has a witness by emptiness of the local wavefront set, and its open cone contains a smaller cone about that direction. Compactness gives a finite subcover ${\mathrm{\Gamma }}_1^{\mathrm{\prime }},\dots ,{\mathrm{\Gamma }}_m^{\mathrm{\prime }}$. For each member of this finite family, take a corresponding witness ${\chi }_j,{\mathrm{\Gamma }}_j$; finite choice suffices. Their witness cutoffs all equal one on a neighborhood of $p$. By [F8], choose one cutoff $\rho$ equal to one near $p$ and supported in the intersection of those neighborhoods. Step 3.2 shows that the single localization $\rho u$ has rapidly decreasing Fourier transform on each ${\mathrm{\Gamma }}_j^{\mathrm{\prime }}$. Since these finitely many cones cover the unit sphere, $\widehat{\rho u}$ is rapidly decreasing in every direction: for each $N$, take the maximum of the finitely many cone bounds outside a bounded ball. Therefore every differentiated inverse Fourier integral is absolutely convergent and defines a smooth function. Fourier inversion [F5] and the pairing interchange [F6], applied after a cutoff equal to one near $\mathrm{supp}(\rho u)$, identify this function with the compactly supported distribution $\rho u$. Hence $u$ is smooth near $p$; as $p$ was arbitrary, it is smooth in every chart.

We first specify two elementary Fourier estimates used with uniform constants. A compact smooth amplitude $a$ has $\mathrm{\mid }\widehat{a}(\eta )\mathrm{\mid }\le C_q(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^{-q}$: for $\mathrm{\mid }\eta \mathrm{\mid }\ge 1$ integrate repeatedly in a coordinate with $\mathrm{\mid }{\eta }_i\mathrm{\mid }\ge \mathrm{\mid }\eta \mathrm{\mid }\mathrm{/}\sqrt{n}$, and use its bounded support for $\mathrm{\mid }\eta \mathrm{\mid }\le 1$. If compact distributions $v_j$ share an order-$m$ bound, [F2] gives $\mathrm{\mid }{\widehat{v}}_j(\eta )\mathrm{\mid }\le C(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^m$ uniformly. Multiplication by $a$ corresponds to convolution with $\widehat{a}$, by [F4]. Uniform rapid decay in a cone transfers to every smaller closed cone under this multiplication: split the convolution into input frequencies in an intermediate regular cone and its complement. In the first region either that frequency is at least half the output size, giving its rapid bound, or the Schwartz factor has size at least half the output size. In the complement, angular separation gives a Schwartz-factor argument bounded below by a positive constant times the sum of both frequency sizes. Taking the decay exponent above $m+n+q+1$ makes each tail integrable and bounds the convolution by $C_q(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^{-q}$. This proves also smooth-factor wavefront containment, with uniform bounds for families. The parameter-pairing lemma justifies the convolution before the estimates; all resulting integrals are absolutely convergent.

Here is the needed nonstationary estimate, including its dependence on the amplitude. On a fixed compact coordinate set let $\mathrm{\Phi }(x;\eta ,\xi )=f(x)\cdot \eta -x\cdot \xi$, put $R=\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\xi \mathrm{\mid }\ge 1$, and suppose $\mathrm{\mid }{\mathrm{\nabla }}_x\mathrm{\Phi }\mathrm{\mid }\ge cR$ throughout the amplitude support. Every fixed $x$ derivative of $\mathrm{\nabla }\mathrm{\Phi }$ is at most $C_{\alpha }R$. Differentiating the quotient shows that every fixed derivative of $b=\mathrm{\nabla }\mathrm{\Phi }\mathrm{/}(2\pi i\mathrm{\mid }\mathrm{\nabla }\mathrm{\Phi }{\mathrm{\mid }}^2)$ is at most $C_{\alpha }^{\mathrm{\prime }}{R}^{-1}$. The operator $L=b\cdot {\mathrm{\nabla }}_x$ satisfies $L{e}^{2\pi i\mathrm{\Phi }}=e^{2\pi i\mathrm{\Phi }}$. Repeated integration by parts gives $\mid \int e^{2\pi i\mathrm{\Phi }}a(x)dx\mid \le C_q{R}^{-q}{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le q}\sup \mathrm{\mid }{\mathrm{\partial }}^{\alpha }a\mathrm{\mid }\mathrm{.}$ Indeed $L^{\ast }a=-{\sum }_i{\mathrm{\partial }}_i(b_{i}a)$, and induction gives a finite sum of derivatives of $a$ through order $q$ times products of $q$ coefficient derivatives, each contributing $R^{-1}$. Compact support removes boundary terms. Constants depend only on finitely many derivatives of $f$, the compact set and $c$, and are uniform for every frequency in the specified region. The same proof with $\xi =0$ applies to $f(x)\cdot \eta$.

We construct the smooth approximants rather than assume a distributional regularization theorem. By [F5] choose $b\ge 0$ compactly supported and equal to one on a neighborhood of a closed box of positive volume inside the unit ball. Rescale its support into the unit ball. By [F6], $0<\int b<\mathrm{\infty }$; put $\rho =b\mathrm{/}\int b$ and ${\rho }_{ϵ}(z)={ϵ}^{-n}\rho (z\mathrm{/}ϵ)$. For a compactly supported distribution $v$, define $v_{ϵ}(y)=⟨v(z),{\rho }_{ϵ}(y-z)\mathrm{\mid }dz\mathrm{\mid }⟩$. It is smooth by [F4], supported in the $ϵ$-neighborhood of $\mathrm{supp}v$, and has common compact support for $0<ϵ\le {ϵ}_0$. Pairing against a test $h$ and using [F4] gives $⟨v_{ϵ},h⟩=⟨v,h_{ϵ}⟩$, where $h_{ϵ}(z)=\int \rho (w)h(z+ϵw)dw$. Every derivative of $h_{ϵ}-h$ tends uniformly to zero on the common compact support, by uniform continuity of the corresponding derivative of $h$. The finite-order bound proves weak convergence and a common order bound, because the averaging bounds every derivative seminorm of $h_{ϵ}$ by the corresponding seminorm of $h$ on a fixed enlarged compact. The same interchange, or Fourier testing, gives $\widehat{v_{ϵ}}(\eta )=\widehat{\rho }(ϵ\eta )\widehat{v}(\eta ),\mathrm{\mid }\widehat{\rho }(ϵ\eta )\mathrm{\mid }\le \int \rho =1.$ Hence all polynomial and regular-cone bounds for $v$ are uniform in $ϵ$. For a smaller local cutoff $a$, first choose $b_0=1$ near its support, inside a wavefront-regularity localization. For sufficiently small $ϵ$, $a({\rho }_{ϵ}\ast v)=a({\rho }_{ϵ}\ast (b_{0}v))$; step 1.1 gives the uniform regular-cone bound. The finitely many initial terms of a sequence ${ϵ}_j\downarrow 0$ are smooth and have their own finite rapid bounds. Compactness and finite subdivision into regular base/cone patches handle any compact cutoff/cone pair disjoint from $\mathrm{WF}(v)$. Thus these approximants are controlled on each smaller patch.

Fix $x_0\in N$ and use one source chart and one target chart about $y_0=f(x_0)$, of positive dimensions for now. On their unit frequency spheres set $K_0=\{\eta :d{f}_{x_0}^T\eta =0\}$. It is compact and disjoint from the compact fiber ${\mathrm{\Gamma }}_{y_0}\cap S^{m-1}$. Choose an open conic neighborhood $V$ of $K_0$ with closure of directions still disjoint from that fiber. Closedness of $\mathrm{\Gamma }$ implies that the same disjointness holds over a sufficiently small target neighborhood: otherwise points of $\mathrm{\Gamma }$ approaching $y_0$, with directions in the compact closure, have a subsequence with a forbidden limit. Outside $V$, compactness gives $\mathrm{\mid }d{f}_{x_0}^T\eta \mathrm{\mid }\ge 2c\mathrm{\mid }\eta \mathrm{\mid }$; shrink the source chart so $\mathrm{\mid }d{f}_x^T\eta \mathrm{\mid }\ge c\mathrm{\mid }\eta \mathrm{\mid }$ there. Shrink again so its image lies in the target neighborhood and choose its target cutoff $\beta$ supported there. If $K_0$ is empty take $V$ empty. Localized Fourier transforms $\widehat{\beta u}$ are rapidly decreasing on $V$: for each compact set of directions in $V$, the regularity witnesses over $\mathrm{supp}\beta$ give a finite base cover and cutoff decomposition; step 1.1 transfers the bounds, and a finite direction cover gives one bound for each order. Choose $V$ initially with closure inside this regular neighborhood, so the bounds hold on its whole set of directions. Only finitely many witnesses are used for each such estimate.

On a compact source test support $H$ inside this small patch, choose $\beta =1$ near $f(H)$ and write $v=\beta u$. For $\omega =a(x)\mathrm{\mid }dx\mathrm{\mid }$ define $T_a(\eta )=\int e^{2\pi if(x)\cdot \eta }a(x)dx,P_f(v)(\omega )=\int \widehat{v}(\eta )T_a(\eta )d\eta \mathrm{.}$ On $V$, $\widehat{v}$ decreases arbitrarily fast and $\mathrm{\mid }{T}_a\mathrm{\mid }\le \mathrm{vol}(H)\sup \mathrm{\mid }a\mathrm{\mid }$. Off $V$, step 1.2 with $\xi =0$ gives $\mathrm{\mid }{T}_a(\eta )\mathrm{\mid }\le C_q(1+\mathrm{\mid }\eta \mathrm{\mid }{)}^{-q}{p}_q(a)$. Take $q>m+m_v+1$, where $m$ is the target dimension and $m_v$ is a finite-order exponent from [F2]. The integrals are absolutely convergent, with $\mathrm{\mid }{P}_f(v)(\omega )\mathrm{\mid }\le C{p}_q(a)$. This proves fixed-support continuity, hence a distribution by [F1]. If $v_j\to v$ is controlled, their common finite-order and regular-cone bounds give the same integrable majorant. Weak convergence gives ${\widehat{v}}_j(\eta )\to \widehat{v}(\eta )$ for each fixed $\eta$, so dominated convergence proves $P_f(v_j)(\omega )\to P_f(v)(\omega )$, with a uniform fixed-support order bound. For smooth $v$, Fourier inversion and absolutely integrable Fubini identify the value with $\int v(f(x))\omega (x)$.

We prove the output cone estimate with uniform bounds, needed also for controlled limits. Fix an output direction ${\xi }_0$ at $x_0$ outside $f^{\ast }\mathrm{\Gamma }$. Shrink the source base patch and its output cone $C$ about ${\xi }_0$. There is a target singular-direction cone $G$ containing every direction of $\mathrm{\Gamma }$ over the target cutoff support such that for $\eta \in G$, $\xi \in C$ and $x$ in the source patch, $\mathrm{\mid }d{f}_x^T\eta -\xi \mathrm{\mid }\ge c_1(\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\xi \mathrm{\mid })\mathrm{.}$ To see this, normalize $\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\xi \mathrm{\mid }=1$. A sequence contradicting the bound has a convergent frequency subsequence and base limit. If the output frequency limit were zero, local transversality from step 2.2 forces the input frequency limit to be zero; if the input limit were zero the equation forces the output limit to be zero. Both contradict normalization. Otherwise the limit is a covector in $f^{\ast }\mathrm{\Gamma }$ with output direction in the chosen excluded closed cone. Closedness and compactness therefore give the separation after shrinking; choose a slightly larger closed input cone still satisfying it. The complement of $G$ is regular for the localized input. To Fourier transform a compact output localization replace $a$ in step 3.1 by $a(x)e^{-2\pi ix\cdot \xi }$. On $G$, apply step 1.2 and the polynomial input bound. Its integral is at most $C_q(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{-q+m+m_v}$ after increasing $q$. Off $G$, use a uniform bound $B$ for $d{f}_x^T$. For $\mathrm{\mid }\eta \mathrm{\mid }\le \mathrm{\mid }\xi \mathrm{\mid }\mathrm{/}(2\max (1,B))$, the phase is nonstationary and the same estimate applies. For the remaining regular input frequencies, the bound on $T_a$ and arbitrary input decay give $C_N(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{-N+m}$ after integration. Taking these exponents arbitrarily large proves rapid output decay on $C$. The estimates use only common input bounds, so they hold uniformly for controlled sequences. The set $f^{\ast }\mathrm{\Gamma }$ is closed locally away from the zero section: transversality bounds $\mathrm{\mid }\eta \mathrm{\mid }$ by a constant times $\mathrm{\mid }d{f}_x^T\eta \mathrm{\mid }$ on $\mathrm{\Gamma }$ over a smaller compact patch, allowing the same compactness argument. Thus the output weak limits and these uniform bounds give exactly the claimed controlled convergence.

The formula is independent of the allowed target cutoff. For two cutoffs equal to one near $f(H)$, first work inside one of the small target neighborhoods of step 2.2. Their difference times $u$ is a compact distribution $w$ zero on a neighborhood of the compact set $f(H)$. Its mollifications from step 2.1 vanish at every $f(x)$, $x\in H$, for sufficiently small $ϵ$, since their supports expand by at most $ϵ$ and the two closed compact sets have positive Euclidean distance. Their ordinary smooth pullbacks therefore pair to zero with $\omega$. The controlled continuity in step 3.1 makes $P_f(w)(\omega )=0$. If the two originally allowed target neighborhoods differ, insert a third cutoff supported in their overlap and equal to one near the image of a smaller source support; compare with it in each neighborhood and use a finite decomposition of $\omega$. This also proves locality and independence of extensions outside the target chart.

Controlled convergence is invariant under a smooth change of coordinates. For completeness, the Fourier estimate is the one for a diffeomorphism, not an assumed arbitrary-map pullback. In source coordinates $y=\psi (z)$, the localized transformed Fourier value pairs $v_j$ with $a(\psi (z))\mathrm{\mid }\det D\psi (z)\mathrm{\mid }{e}^{-2\pi i\psi (z)\cdot \eta }$. Insert a cutoff equal to one on this compact support and use Fourier inversion [F4]. The resulting phase is $z\cdot \zeta -\psi (z)\cdot \eta$. In the near-stationary region $\zeta$ is comparable to $\eta$ and lies in a regular source cone whenever the transformed compact base/output cone avoids the transported $\mathrm{\Gamma }$; there use the uniform regular bounds and a polynomial volume estimate. On the complementary region $\mathrm{\mid }\zeta -D\psi (z{)}^T\eta \mathrm{\mid }\ge c(\mathrm{\mid }\zeta \mathrm{\mid }+\mathrm{\mid }\eta \mathrm{\mid })$; step 1.2 and the uniform polynomial bound give arbitrary rapid decay after integration. This is precisely the chart-invariance estimate with constants uniform in $j$. Common finite-order bounds transfer by the finite chain/product rules for compactly supported test densities, and weak convergence transfers because diffeomorphism pullback is already defined by the density transpose. Hence the same smooth approximants from step 2.1, expressed in either target chart, are controlled. Both local candidates in step 3.1 are their limits and equal the same ordinary smooth pullback at every stage. They are therefore equal. A change of source chart similarly gives the same smooth integrals by the absolute-Jacobian substitution in [F6], followed by the controlled limit. This proves coordinate covariance of the constructed scalar distribution and compatibility with the previously defined diffeomorphism pullback.

We now glue; no assumption that $f(\mathrm{supp}\omega )$ fits one chart is made. Cover the compact support of a test density by finitely many small source patches of step 2.2. Shrink the covering patches still covering that support, take compact smooth cutoffs $b_i$ supported in the original patches and equal to one on the smaller ones, and put $s={\sum }_i{b}_i$. It is positive near the test support. With another compact cutoff $h=1$ there and supported where $s>0$, set $a_i=h{b}_i\mathrm{/}s$ there and zero elsewhere. They are smooth, subordinate, and sum to one near the test support. Define $⟨f^{\ast }u,\omega ⟩={\sum }_i{P}_{f,i}(u)(a_i\omega )$. On an overlap the two terms are equal: shrink around each point to a common source patch and a target chart inside the two target-chart overlap, use step 4.2 for cutoffs and step 5.1 for coordinates, and decompose a compact overlap test into finitely many such patches. Two partitions now give the same answer by the double sum on $a_i{a}_j^{\mathrm{\prime }}\omega$. This proves linearity, independence and compatibility with restriction. For each fixed compact support choose this finite cover once. Each local pairing has a finite seminorm estimate by step 3.1; multiplication by the fixed $a_i$ is bounded by finitely many compact derivative seminorms. Their finite sum is continuous on that fixed-support space, and the test-density topology in [F1] gives a global distribution. Local estimates in step 4.1 give its wavefront bound and controlled sequential continuity.

If a source or target coordinate dimension is zero, every local distribution there is a scalar function, and the relevant local map is an ordinary smooth map from or to a point. For a zero-dimensional source, transversality forces the target distribution to be smooth near its image point, by [F3], so evaluation is defined; for a zero-dimensional target its local scalar pulls back as a constant. The finite-test gluing above handles zero-dimensional components, and an empty source has only the zero test space and its unique distribution. Zero inputs pull back to zero. Nonproper $f$ has required no extra hypothesis: each construction used only a compact source test support and its compact image. Finally any other rule agreeing on smooth inputs and obeying the stated controlled sequential continuity agrees on each small patch by the actual approximants of step 2.1, hence agrees globally by step 6.1. All coverings and comparisons for one test use finite choice only; [A1] enters exactly through the declared Fourier and Lebesgue suppliers.

In a pair of coordinate charts, localize compactly supported $u$ and $v$. By [F5], the localized tensor transform factors as $\widehat{u\otimes v}(\xi ,\zeta )=\widehat{u}(\xi )\widehat{v}(\zeta )$. If both covectors are regular, each factor decreases faster than every power on a conic neighborhood; if only one is singular, rapid decay in the regular block controls the other polynomial factor whenever that regular block has nonzero central covector. On a sufficiently small cone about such a full covector, the nonzero block frequency is comparable to the total frequency, so its decay can be taken above every polynomial order and the fixed dimension. The cases with one central covector zero use the other nonzero regular block in the same way. The finite-order bounds [F3] control the remaining factor polynomially. Thus $\mathrm{WF}(u\otimes v)\subseteq (\mathrm{WF}(u)\times \mathrm{WF}(v))\cup (\mathrm{WF}(u)\times 0_M)\cup (0_M\times \mathrm{WF}(v))$. Finite chart covers and cutoffs [F7] give the same inclusion on manifolds.