Flat hyperplanes do not have spherical stationary-phase decay

The transform of the flat measure. Since $\mu$ is carried by $\{{\omega }_n=0\}\cong [-1,1{]}^{n-1}$ with density $1$, [F1] gives $\check{\mu }(x)={\int }_{[-1,1{]}^{n-1}}{e}^{2\pi i{x}^{\mathrm{\prime }}\cdot {\omega }^{\mathrm{\prime }}}d{\omega }^{\mathrm{\prime }}$: the variable $x_n$ does not appear. By Fubini over the product $[-1,1{]}^{n-1}$ and the one-dimensional evaluation [F2], $\check{\mu }(x)={\prod }_{j<n}{\int }_{-1}^1{e}^{2\pi i{x}_{j}t}dt={\prod }_{j<n}\frac{\sin 2\pi x_j}{\pi x_j},$ with each factor read as its continuous value $2$ at $x_j=0$.

No decay along the normal. Setting $x^{\mathrm{\prime }}=0$ gives $\check{\mu }(0,x_n)={\prod }_{j<n}2=2^{n-1}$ for every $x_n\in \mathbb{R}$, since the product is independent of $x_n$. Along the normal line $x^{\mathrm{\prime }}=0$ the function is the nonzero constant $2^{n-1}$, so for no constant $C$ can $\mathrm{\mid }\check{\mu }(x)\mathrm{\mid }\le C(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-(n-1)\mathrm{/}2}$ hold for all $x$: as $\mathrm{\mid }{x}_n\mathrm{\mid }\to \mathrm{\infty }$ the right-hand side tends to $0$ while the left remains $2^{n-1}$. This refutes the curvature-free statement, and it shows that the hypothesis in [F3] is necessary.

The sinc product is continuous and positive at $x^{\mathrm{\prime }}=0$, so its modulus is bounded below on a tangential ball of positive measure. It is independent of $x_n$; Tonelli on that ball times $\mathbb{R}$ gives $\int \mathrm{\mid }Eg{\mathrm{\mid }}^q=\mathrm{\infty }$ for every $1\le q<\mathrm{\infty }$, while $g\in L^2(\mathrm{\Sigma })$. To test the smooth-localization hypothesis itself, take a nonnegative nonzero $b\in C_c^{\mathrm{\infty }}({\mathbb{R}}^{n-1})$. Then $\int e^{2\pi i{x}^{\mathrm{\prime }}\cdot {\omega }^{\mathrm{\prime }}}b({\omega }^{\mathrm{\prime }})d{\omega }^{\mathrm{\prime }}$ is independent of $x_n$ and equals $\int b>0$ at $x^{\mathrm{\prime }}=0$, so it also fails the decay estimate. It is the restriction of an ambient smooth cutoff times $b$, and hence is an allowed smooth localized measure on the flat graph. The compact-surface conclusion also needs curvature: a sphere can be modified on its upper graph by replacing $\sqrt{R^2-\mathrm{\mid }y{\mathrm{\mid }}^2}$ with $\chi (y)R+(1-\chi (y))\sqrt{R^2-\mathrm{\mid }y{\mathrm{\mid }}^2}$, where $\chi =1$ on a small ball and vanishes outside a larger ball strictly inside $\mathrm{\mid }y\mathrm{\mid }<R$. The resulting compact smooth embedded hypersurface has a flat open patch. A nonzero smooth density supported in that patch gives the same normal-coordinate independence and rules out every finite-$q$ extension estimate.

Conclusion. Steps 1.1–2.2 exhibit the flat localization $d\mu ={\mathbf{1}}_{[-1,1{]}^{n-1}}d{\omega }^{\mathrm{\prime }}$ whose transform does not decay in the normal direction and whose extension fails every finite-$q$ bound; in particular the curvature hypothesis in the curved-patch decay and in the compact-hypersurface corollary cannot be removed.