Analytic cutoff sequence and local FBI invariance

Fix one real-analytic chart and two smooth cutoffs $\chi ,\tilde{\chi }$ that both equal one on a neighborhood of the base point. Choose a smaller center neighborhood $U_0$ and a compact support neighborhood so that $\mathrm{\mid }x-y\mathrm{\mid }\ge d>0$ whenever $x\in U_0$ and $y\in \mathrm{supp}(\chi -\tilde{\chi })$. By continuity in the fixed-support topology [F3], in either distributional type the pairing is bounded by $C_0$ times the maximum of finitely many derivatives of its test object; a coordinate density factor, when present, has bounded derivatives on the compact chart support. Those derivatives of $(\chi -\tilde{\chi })e^{i\lambda (x-y)\cdot \omega -\lambda \mathrm{\mid }x-y{\mathrm{\mid }}^2\mathrm{/}2}$ are bounded by $C_1(1+\lambda {)}^m{e}^{-\lambda d^2\mathrm{/}2}$ uniformly for $x\in U_0$, directions near ${\omega }_0$, and $\lambda \ge 1$. Since a polynomial times $e^{-\lambda d^2\mathrm{/}2}$ is bounded by $C_2{e}^{-c\lambda }$ for any fixed $0<c<d^2\mathrm{/}2$, replacing one fixed-chart cutoff by the other changes the FBI packet by an exponentially small term.

Let $\kappa$ be an analytic chart change. After shrinking to nested compact neighborhoods, $\mathrm{\mid }\kappa (q)-\kappa (y)\mathrm{\mid }\ge c\mathrm{\mid }q-y\mathrm{\mid }$ and $D\kappa (q)$ is invertible uniformly. The pulled-back packet has phase $\varphi (y,q,\omega )=\omega \cdot (\kappa (q)-\kappa (y))+\frac{i}{2}\mathrm{\mid }\kappa (q)-\kappa (y){\mathrm{\mid }}^2\mathrm{.}$ It is holomorphic after complexification, vanishes at $y=q$, and has derivative $-D\kappa (q{)}^T\omega$ there. First [F6] allows $\omega$ to range over a bounded open ball about its nonzero reference direction: its reference phase $(q-y)\cdot \omega +i\mathrm{\mid }\omega \mathrm{\mid }\mathrm{\mid }q-y{\mathrm{\mid }}^2\mathrm{/}2$ is exactly the unit-direction packet with parameter $\lambda \mathrm{\mid }\omega \mathrm{\mid }$, and [F6] equates this phase to the Gaussian of width $\lambda$ on that open ball. Apply the same reference criterion in the new chart before pulling its packets back. We may therefore reparameterize the open frequency ball by $p=D\kappa (q{)}^T\omega$; this is an analytic local diffeomorphism of open frequency parameters and is precisely the cotangent rule. The phase is now normalized and positive in the sense of [F6]. The Jacobian amplitude for scalar distributions is analytic and nonzero: the sign of the real determinant is constant on the smaller connected chart, so its absolute value has an analytic continuation. For distributional densities the amplitude is $1$.

Thus the exponential-decay criterion is independent of the fixed-chart cutoff. The finite-order sequence [F2] supplies the uniform localizers required when a distribution is restricted to a smaller chart; it avoids any use of compactly supported analytic bumps. For a distributional density $V$ in the fixed chart, define the scalar distribution $u_V$ by $⟨u_V,\phi \mathrm{\mid }dy\mathrm{\mid }⟩=⟨V,\phi ⟩$. Continuity follows from [F3], and the vanishing-open sets of $u_V$ and $V$ agree because $\mathrm{\mid }dy\mathrm{\mid }$ is nowhere zero, so their supports agree. The packet pairings agree by definition. Hence [F4]–[F5] give the local Euclidean tempered and smooth-function realizations for either distributional type. This proves the cutoff and localization clauses without a choice axiom.

Apply [F6] to the standard and pulled-back packets on the smaller neighborhoods. It supplies actual transition representations in both directions, with polynomially bounded transition kernels and errors exponentially small in every fixed derivative; the finite distributional order from [F3] therefore preserves their exponential size after pairing. The known exponential estimate on the source parameter neighborhood transfers to the target one. Swapping the two families gives the converse. Step 1.1 removes all fixed cutoff choices, and step 2.1 covers both distributional conventions. This proves every invariance clause in the statement, with the exact cotangent transformation from step 1.2.