Decay of a localized measure on a curved graph patch

Put $d=n-1$, $b=a\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2}$ and $K=\mathrm{supp}a$. Choose a compact neighbourhood $K^{+}$ of $K$ inside $U$ on which $D^{2}h$ is invertible. This exists by continuity and a finite cover of $K$. Every derivative of $h$ needed below is bounded there. For $x=\rho \omega$, $\rho \ge 1$ and $\omega \in S^{n-1}$, the phase is ${\psi }_{\omega }(y)={\omega }^{\mathrm{\prime }}\cdot y+{\omega }_{n}h(y)$ and the integral is $\int e^{2\pi i\rho {\psi }_{\omega }}b$. Its gradient is ${\omega }^{\mathrm{\prime }}+{\omega }_n\mathrm{\nabla }h$. At a zero in $K^{+}$, $\mathrm{\mid }{\omega }_n\mathrm{\mid }=(1+\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$, so the Hessian ${\omega }_n{D}^{2}h$ is invertible with uniformly bounded inverse.

Fix a direction ${\omega }_0$. Its zeros in $K^{+}$ are isolated by the inverse theorem in [F2]. Only finitely many lie in a smaller compact neighbourhood of $K$: otherwise compactness gives an accumulating zero in $K^{+}$, contradicting local invertibility. Surround these finitely many zeros by disjoint small balls compactly contained in $K^{+}$, on which $\mathrm{\nabla }h$ is injective; choose smaller concentric balls around the zeros. Every remaining point of $K$ has nonzero phase gradient at ${\omega }_0$. A fixed finite smooth partition on a neighbourhood of $K$ therefore splits $b$ into amplitudes supported either in these zero balls or on a compact set where $\mathrm{\mid }\mathrm{\nabla }{\psi }_{{\omega }_0}\mathrm{\mid }\ge c>0$.

Shrink a neighbourhood $V$ of ${\omega }_0$ in the direction sphere. On the nonstationary support, continuity keeps the gradient at least $c\mathrm{/}2$. For each zero ball, the implicit theorem provides a smooth critical point $z(\omega )$ remaining in its smaller ball for $\omega \in V$. Injectivity of $\mathrm{\nabla }h$ and ${\omega }_n\mathrm{\ne }0$ ensure it is the only critical point in the larger ball. By shrinking the ball and $V$, Taylor's formula makes $\mathrm{\mid }\mathrm{\nabla }{\psi }_{\omega }(y)\mathrm{\mid }\ge c_0\mathrm{\mid }y-z(\omega )\mathrm{\mid }$ near that point uniformly: subtract the gradient at $z$ and use uniform closeness of the Hessian to its invertible value at $({\omega }_0,z({\omega }_0))$. On the compact remainder of the ball the gradient stays bounded below after further shrinking $V$. All required derivatives and Hessian inverses are uniformly bounded.

Apply [F1] on these supports. The nonstationary amplitudes give $O({\rho }^{-N})$ uniformly on $V$. The zero-ball amplitudes give $O({\rho }^{-d\mathrm{/}2})$ uniformly, even when the critical point lies outside the amplitude support: add a fixed smooth bump supported in that ball, equal to one on its smaller ball and multiplied by a constant larger than the amplitude bound, then subtract the same bump. Each of the two new amplitudes has the critical point in the interior of its support and uniformly bounded derivatives, so the stated stationary estimate applies to each; the local proof uses only the phase on that ball. Thus the original amplitude has the same bound by subtraction. This also handles critical points entering or leaving the original support.

The neighbourhoods $V$ constructed for each direction cover the compact sphere, so finitely many suffice. Taking the maximum of their finite constants gives $\mathrm{\mid }\check{\mu }(x)\mathrm{\mid }\le C\mathrm{\mid }x{\mathrm{\mid }}^{-d\mathrm{/}2}$ for $\mathrm{\mid }x\mathrm{\mid }\ge 1$. For every $x$, the pointwise estimate $\mathrm{\mid }\check{\mu }(x)\mathrm{\mid }\le \int \mathrm{\mid }b(y)\mathrm{\mid }dy<\mathrm{\infty }$ follows from unit modulus of the exponential and bounded compact support. Combining the two bounds yields $C_a(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-d\mathrm{/}2}$ after increasing $C_a$. No constancy of the number of critical points over the whole sphere is asserted or used.

For the general clause, $K={\mathrm{supp}}_S(\phi {\mathrm{\mid }}_S)$ is compact by the explicit hypothesis. Apply [F2] to $K$, and [F3] to its graph charts; shrink the charts to retain nondegenerate Hessians on the compact supports of the localized weights. The graph amplitudes $a_j=({\chi }_j\phi )\circ X_j$ are smooth and compactly supported in their parameter domains. The graph measure formula in [F4] writes $\phi \sigma$ as their finite sum. A rigid motion rotates the frequency and contributes only a scalar exponential of modulus one, so step 5.1 applies without changing $\mathrm{\mid }x\mathrm{\mid }$. Summing proves the asserted general decay.