Product and pullback under wavefront transversality

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.

Apply [F8] with $\mathrm{\Gamma }=\mathrm{WF}(u)$ and the stated map $f$. Its transversality hypothesis is exactly $\mathrm{WF}(u)\cap N_f=\mathrm{\varnothing }$. At each $x_0$ it first chooses one small source chart and one target chart about $f(x_0)$ with a uniform cone separation on those patches. Only on such a patch is the local value $\int \widehat{\beta u}(\eta )T_{\omega }(\eta )d\eta$ used, with $T_{\omega }(\eta )=\int e^{2\pi if(x)\cdot \eta }\omega (x)$ and $\beta =1$ near the compact test image. The complete proof in [F8] establishes the finite-seminorm bound, controlled smooth limits and the full output wavefront estimate, proves independence of $\beta$ by support-separated mollifications, and proves coordinate covariance by transforming the same controlled sequence. A finite source partition and a double-partition refinement then give overlap agreement and global gluing even when the full test image spans target charts. Thus $f^{\ast }u$ is the invariant distribution in the Statement, with its stated wavefront inclusion; no single global normal cone for an unrelated compact image is assumed.

For the diagonal $\mathrm{\Delta }:M\to M\times M$, $d{\mathrm{\Delta }}_x^T(\eta ,\zeta )=\eta +\zeta$ and $N_{\mathrm{\Delta }}=\{(x,x;\eta ,-\eta ):\eta \mathrm{\ne }0\}$. By step 1.1, $\mathrm{WF}(u\otimes v)$ can meet $N_{\mathrm{\Delta }}$ only if $(x,\eta )\in \mathrm{WF}(u)$ and $(x,-\eta )\in \mathrm{WF}(v)$ for a common nonzero $\eta$. The hypothesis excludes this, so step 1.2 defines ${\mathrm{\Delta }}^{\ast }(u\otimes v)$. When both factors are smooth, the Fourier pullback formula in step 1.2 gives their ordinary pointwise product.

First shrink about one base point so that the projections of the two closed singular cones still exclude opposite directions, by closedness and compactness of normalized covectors, and place both input cutoffs inside that small patch. For smooth compact approximants, Fourier inversion and the absolutely integrable parameter interchange in [F4] give the Fourier-convolution expression; controlled mollifications from [F8], the uniform version of step 1.1, and diagonal pullback continuity preserve it in the limit, giving the local expression $\widehat{uv}(\xi )=\int \widehat{u}(\eta )\widehat{v}(\xi -\eta )d\eta$. The no-opposite-covector hypothesis separates the closed conic directions of $\widehat{u}$ and $\widehat{v}$ from cancellation. On a compact base patch their unit directions have a positive angular separation, so when both frequencies lie in their singular cones, $\mathrm{\mid }\eta \mathrm{\mid }+\mathrm{\mid }\xi -\eta \mathrm{\mid }\le C\mathrm{\mid }\xi \mathrm{\mid }$. In the tail $\mathrm{\mid }\eta \mathrm{\mid }>2\mathrm{\mid }\xi \mathrm{\mid }$, the two directions are nearly opposite, hence at least one factor is outside its singular cone and is rapidly decreasing; [F3] bounds the other polynomially. The integral is absolutely convergent and has polynomial growth, using [F6]. If $\xi$ lies outside $\mathrm{WF}(u)\cup \mathrm{WF}(v)\cup (\mathrm{WF}(u)+\mathrm{WF}(v))$, choose a smaller cone still outside that union; the same split makes at least one factor rapidly decreasing in every part of the convolution, proving rapid decay there. This proves the displayed wavefront inclusion and shows that the convolution product is exactly the diagonal pullback from step 2.1. The same controlled limits identify this convolution with the canonical diagonal pullback. Cutoff/coordinate independence and the finite gluing in [F8] therefore glue these local products; no equality of arbitrarily defined singular products is presumed.

For the projection $\pi$ and smooth positive density $\nu$, define ${\pi }_{\ast ,\nu }t$ by pairing $t$ with $\chi ({\pi }^{\ast }\omega \otimes \nu )$, where $\chi =1$ near $\mathrm{supp}t$. A change of cutoff pairs $t$ with a test density zero near its support, so the value is independent of $\chi$; the support lies in the compact set $\pi (\mathrm{supp}t)$. In a product chart, multiplication by the smooth coefficient of $\nu$ does not enlarge the wavefront set by step 3.1, and the localized Fourier transform in the base variables is the full transform of $\chi t$ evaluated at vertical frequency zero. Therefore rapid decay can fail only if $(x,y;\eta ,0)\in \mathrm{WF}(t)$ for some $y$, giving the stated estimate. For each excluded output direction, compactness of the vertical support gives finitely many product-chart localizations with one smaller base cone on which the localized transforms decay rapidly at vertical frequency zero; sum their bounds. Finite chart covers and cutoffs [F7] therefore give the manifold statement.