Analytic stationary phase with optimal truncation

Choose $r>0$ so that ${\chi }_N=1$ on $B_{2r}(0)$ for every $N$, the phase and amplitude are holomorphic near ${\bar{B}}_{2r}(0)$, and the stated positivity holds. The positive-phase hypothesis and compactness of $\mathrm{supp}a\setminus B_r(0)$ give $\mathrm{Im}(\mathrm{\Phi }-{\mathrm{\Phi }}_0)\ge d>0$ there. Since $a$ is bounded and ${\chi }_N$ is bounded by one, the absolute value of that contribution after multiplication by $e^{-i\lambda {\mathrm{\Phi }}_0}$ is at most $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }U\mathrm{\mid }{e}^{-\lambda d}$, uniformly in $N$. Thus only the integral over the fixed local ball affects the power series; the complement contributes $O(e^{-d\lambda })$.

Put $\varphi =-i(\mathrm{\Phi }-{\mathrm{\Phi }}_0)$, so $\mathrm{Re}\varphi =\mathrm{Im}(\mathrm{\Phi }-{\mathrm{\Phi }}_0)$ and ${\varphi }^{\mathrm{\prime }\mathrm{\prime }}(0)=H$. Apply [F1] to obtain a holomorphic Morse coordinate $w=\mathrm{\Psi }(x)$ with $\varphi (x)=\frac{1}{2}w\cdot w$. The transformed real ball is a local contour $\mathrm{\Gamma }$ through $0$. The transformed amplitude $b(w)=a({\mathrm{\Psi }}^{-1}(w))\det D{\mathrm{\Psi }}^{-1}(w)$ is holomorphic in a fixed polydisc and has coefficients bounded by $M{R}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ for some $M,R>0$ by [F3]–[F5].

Differentiating the Morse identity in [F1] twice at $0$ gives $A^{T}A=H$, where $A=D\mathrm{\Psi }(0)$. Taylor's formula gives $\mathrm{Re}(\varphi (tv))=\frac{1}{2}{t}^2{v}^T(\mathrm{Re}H)v+o(t^2)$; comparing with $\mathrm{Re}\varphi (tv)\ge c{t}^2\mathrm{\mid }v{\mathrm{\mid }}^2$ and dividing by $t^2\mathrm{/}2$ as $t\to 0$ yields $v^T(\mathrm{Re}H)v\ge 2c\mathrm{\mid }v{\mathrm{\mid }}^2$. For a real vector $v$, write $Av=u+iw$ with $u,w\in {\mathbb{R}}^n$. Then $\mathrm{\mid }u{\mathrm{\mid }}^2-\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{Re}(v^{T}Hv)>0$ when $v\mathrm{\ne }0$. In particular, the real projection on $A{\mathbb{R}}^n$ is injective, hence bijective by dimension, and $A{\mathbb{R}}^n=\{u+iKu:u\in {\mathbb{R}}^n\}$ for a real linear map $K$. For nonzero $u$, applying the strict inequality to $v=A^{-1}(u+iKu)$ gives $\mathrm{\mid }Ku\mathrm{\mid }<\mathrm{\mid }u\mathrm{\mid }$; compactness of the unit sphere gives $\mathrm{\parallel }K\mathrm{\parallel }<1$. The real inverse theorem [F6] makes the nonlinear contour $\mathrm{\Gamma }$ a graph $u+i(Ku+r(u))$, with $r(u)=O(\mathrm{\mid }u{\mathrm{\mid }}^2)$. For sufficiently small $r_0$, the homotopy graphs $u+i(Ku+tr(u))$ remain in the holomorphic chart and satisfy $\mathrm{Re}(w\cdot w)\ge c_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{u}\mathrm{r}}\mathrm{\mid }u{\mathrm{\mid }}^2$ for $\mathrm{\mid }u\mathrm{\mid }\le r_0$, because the linear margin $\mathrm{\mid }u{\mathrm{\mid }}^2-\mathrm{\mid }Ku{\mathrm{\mid }}^2\ge (1-\mathrm{\parallel }K{\mathrm{\parallel }}^2)\mathrm{\mid }u{\mathrm{\mid }}^2$ dominates the $O(\mathrm{\mid }u{\mathrm{\mid }}^3)$ error. The homotopy $u+itKu$, $0\le t\le 1$, also has $\mathrm{Re}(w\cdot w)=\mathrm{\mid }u{\mathrm{\mid }}^2-t^2\mathrm{\mid }Ku{\mathrm{\mid }}^2\ge (1-\mathrm{\parallel }K{\mathrm{\parallel }}^2)\mathrm{\mid }u{\mathrm{\mid }}^2$. On the fixed outer boundary these bounds, together with boundedness of the transformed amplitude and Jacobian, make the integrals $O(e^{-c_1\lambda })$. [F7] and the contour statement in [F1] therefore deform $\mathrm{\Gamma }$ to the real Gaussian contour, with only that exponential boundary term.

Truncate the Taylor series after total degrees below $2N$. On a smaller ball $\mathrm{\mid }w\mathrm{\mid }\le r_0<R$, the Cauchy bound [F5] and a geometric-series estimate give $\mid b(w)-{\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }<2N}{b}_{\alpha }{w}^{\alpha }\mid \le C_2^{N+1}\mathrm{\mid }w{\mathrm{\mid }}^{2N}\mathrm{.}$ Integrating this remainder against $e^{-\lambda \mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}2}$ and using $z=\sqrt{\lambda }w$ bounds it by $C_3^{N+1}{\lambda }^{-N-n\mathrm{/}2}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }z{\mathrm{\mid }}^{2N}{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}dz\le C_4^{N+1}N!{\lambda }^{-N-n\mathrm{/}2}\mathrm{.}$ For the last inequality, convexity gives $({\sum }_{j=1}^n{z}_j^2{)}^N\le n^{N-1}{\sum }_{j=1}^n\mathrm{\mid }{z}_j{\mathrm{\mid }}^{2N}$. Fubini factors each summand into one-dimensional Gaussian moments; integration by parts gives $M_N=(2N-1)M_{N-1}$ with $M_0=\sqrt{2\pi }$, since the polynomial times Gaussian boundary term tends to zero. Hence $M_N=(2N-1)!!\sqrt{2\pi }\le 2^{N}N!\sqrt{2\pi }$, and the displayed integral is at most $(2n{)}^N(2\pi {)}^{n\mathrm{/}2}N!$. Absorb this fixed-dimensional factor into $C_4^{N+1}$. The extension of the Taylor polynomial $P_{<2N}(w)={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }<2N}{b}_{\alpha }{w}^{\alpha }$ from $\mathrm{\mid }w\mathrm{\mid }\le r_0$ to the full real Gaussian integral needs a separate estimate uniform in $N$. For $\mathrm{\mid }w\mathrm{\mid }\ge r_0$ and $k<2N$, $\mathrm{\mid }w{\mathrm{\mid }}^k\le r_0^{k-2N}\mathrm{\mid }w{\mathrm{\mid }}^{2N}$. Cauchy's coefficient bound and the at most exponential count of degree-$k$ multi-indices give $D_N:={\sum }_{k<2N}{\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=k}\mathrm{\mid }{b}_{\alpha }\mathrm{\mid }{r}_0^{k-2N}\le C_5^{N+1}$, since $r_0,R$ are fixed positive constants and the sum has $2N$ terms after grouping degrees. Consequently ${\int }_{\mathrm{\mid }w\mathrm{\mid }\ge r_0}\mathrm{\mid }{P}_{<2N}(w)\mathrm{\mid }{e}^{-\lambda \mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}2}dw\le C_5^{N+1}{\lambda }^{-N-n\mathrm{/}2}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }z{\mathrm{\mid }}^{2N}{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}dz\le C_6^{N+1}N!{\lambda }^{-N-n\mathrm{/}2}$. Thus the polynomial extension tail is controlled by the factorial remainder for every $N$, rather than by an exponential constant independent of $N$. The actual amplitude away from the local critical region and the contour boundary still have the exponential estimates from step 1.1 and [F1]; no extension of that amplitude outside its holomorphy neighborhood is used. Combining these estimates proves the stated normalized error for every $N$.

Expand $b(w)={\sum }_{\alpha }{b}_{\alpha }{w}^{\alpha }$ and group by total degree. Odd monomials integrate to zero over ${\mathbb{R}}^n$ because the Gaussian and domain are invariant under $w\mapsto -w$. Each monomial of degree $2j$ contributes a constant times ${\lambda }^{-n\mathrm{/}2-j}$ by the change of variables $z=\sqrt{\lambda }w$. In one dimension, integration by parts gives $I_k=\int t^{2k}{e}^{-t^2\mathrm{/}2}dt=(2k-1)I_{k-1}$ and $I_0=\sqrt{2\pi }$, hence $I_k=(2k-1)!!\sqrt{2\pi }$. Products give the $n$-dimensional moments. If $\alpha =2k$ and $\mathrm{\mid }k\mathrm{\mid }=j$, then ${\prod }_l(2k_l-1)!!\le (2j-1)!!\le 2^{j}j!$; all monomials with an odd coordinate vanish. After scaling, every degree-$2j$ moment is therefore at most $C_0^{j+1}j!{\lambda }^{-j}$. The number of degree-$2j$ multi-indices is at most $C_n^{j+1}$ and $\mathrm{\mid }{b}_{\alpha }\mathrm{\mid }\le M{R}^{-2j}$, so the coefficient of ${\lambda }^{-j}$ is bounded by $C_1^{j+1}j!$. The Jacobian at zero contributes $(\det H{)}^{-1\mathrm{/}2}$ on the chosen branch, and the degree-zero coefficient is $a(0)$.

For $N=\lfloor ϵ\lambda \rfloor$ and sufficiently large $\lambda$ so that $N\ge ϵ\lambda \mathrm{/}2\ge 1$, the factorial term is at most $C^{N+1}{\lambda }^{-n\mathrm{/}2}(N\mathrm{/}\lambda {)}^N$ because $N!\le N^N$. Choose $ϵ>0$ with $Cϵ<1\mathrm{/}2$. Since $N\le ϵ\lambda$, the displayed term is at most $C^{N+1}{\lambda }^{-n\mathrm{/}2}{ϵ}^N$, which is bounded by $C^{\mathrm{\prime }}{e}^{-{\delta }_2\lambda }$ because $N\ge ϵ\lambda \mathrm{/}2$ and $2^{-N}\le e^{-(ϵ\log 2)\lambda \mathrm{/}2}$. Combining this with the exponential term in step 3.2 proves optimal truncation of the normalized remainder at order proportional to $\lambda$, with all stationary-phase coefficients retained. For the absolute remainder multiply this bound by $\mathrm{\mid }{e}^{i\lambda {\mathrm{\Phi }}_0}\mathrm{\mid }=e^{-\lambda \mathrm{Im}{\mathrm{\Phi }}_0}$; no sign restriction on $\mathrm{Im}{\mathrm{\Phi }}_0$ is imposed.