Local analytic elliptic packet reconstruction

We retain the formal inverse algebra needed to identify the external operator construction [F9]. For a formal holomorphic symbol $p={\sum }_{j\ge 0}{h}^j{p}_j(x,\xi )$, with $\mathrm{\mid }{p}_j\mathrm{\mid }\le C^{j+1}j!$ on a fixed polydisc, define $p\mathrm{\#}q={\sum }_{j,k,\gamma }\frac{h^{j+k+\mathrm{\mid }\gamma \mathrm{\mid }}}{i^{\mathrm{\mid }\gamma \mathrm{\mid }}\gamma !}({\mathrm{\partial }}_{\xi }^{\gamma }{p}_j)({\mathrm{\partial }}_x^{\gamma }{q}_k)\mathrm{.}$ There are finitely many terms in each degree. Associativity follows directly by applying the product rule to the two triple products: splitting a multi-index $\gamma =\mu +\nu$ contributes $\gamma !\mathrm{/}(\mu !\nu !)$, which cancels its denominator and leaves identical terms. The map $p\mapsto P=p(x,\xi +h{D}_x)$, with $D_x={\mathrm{\partial }}_x\mathrm{/}i$, respects this product; expansion by the product rule gives the displayed formula.

The actual contour realization is [F8], with $\rho =-\mathrm{Im}S$ and $h={\lambda }^{-1}$. In fixed-offset coordinates $y=x+w$, its oriented volume and prefactor give the explicit formula

Here $dL$ is Euclidean $2n$-dimensional measure, the phase-space cycle and its orientation are exactly those of [F8], and $R,r$ are fixed within the common holomorphy neighborhood. The Gaussian bound and amplitude-$1$ identity are the specified operator theorem [F8]; equivalently, for amplitude $1$ the holomorphic factor $e^{-iw\cdot S_x(x)\mathrm{/}h}v(x+w;h)$ has constant Taylor coefficient $v(x;h)$ and all positive-degree holomorphic monomials have zero angular integral. The remaining Gaussian tail is exponentially small in the weighted germ. For this pluriharmonic weight the fixed-offset formula is holomorphic in $x$; it also specifies precisely the previously implicit cycle and measure. The fixed-derivative real-localization bounds used later are [F10], rather than an assertion obtained solely from this formal identity.

Here are estimates for this formal algebra that also control its inverse. Use nested complex $x$ neighborhoods ${\mathrm{\Omega }}_t$, $0<t<t_0<1$, whose boundary separation is at least $t-s$, and keep $\xi$ in a smaller fixed polydisc. Write $P=\sum h^k{P}_k$, where $P_k$ has differential order at most $k$. Cauchy's inequality, first in the fixed $\xi$ polydisc and then in $x$, gives $\mathrm{\parallel }{P}_k{\mathrm{\parallel }}_{t,s}\le C^{k+1}{k}^k(t-s{)}^{-k},0<s<t<t_0\mathrm{.}$ Indeed a term with $j+\mathrm{\mid }\gamma \mathrm{\mid }=k$ costs at most $C^{k+1}j!\gamma !(t-s{)}^{-\mathrm{\mid }\gamma \mathrm{\mid }}$; $j!\gamma !\le k!\le k^k$, $t-s<1$, and the number of terms is at most exponential in $k$. For $k=0$ use $0^0=1$. Let $f_k(P)$ be the least admissible constant in $\mathrm{\parallel }{P}_k{\mathrm{\parallel }}_{t,s}\le f_k(P)k^k(t-s{)}^{-k}$. If $R=PQ$, choose the intermediate radius with $r-s=j(t-s)\mathrm{/}(j+k)$ for the term $P_j{Q}_k$. Then $f_{\mathrm{\ell }}(PQ)\le {\sum }_{j+k=\mathrm{\ell }}{f}_j(P)f_k(Q)\mathrm{.}$ The endpoint cases use the same estimate without an intermediate loss. Consequently $\mathrm{\parallel }P{\mathrm{\parallel }}_{\rho }=\sum {\rho }^k{f}_k(P)$ is submultiplicative whenever finite. Conversely $p_k=P_{k}1$, evaluated with a fixed radius loss, satisfies $\mathrm{\mid }{p}_k\mathrm{\mid }\le C^{k+1}{k}^k\le (eC{)}^{k+1}k!$; the last inequality follows by integrating $\log t$ from $1$ to $k$. These estimates are on one fixed smaller neighborhood, independent of $k$.

If $p_0\mathrm{\ne }0$, shrink so $1\mathrm{/}{p}_0$ is bounded holomorphic. The formal product $p\mathrm{\#}(1\mathrm{/}{p}_0)=1-r$ has $r_0=0$ and finite $\mathrm{\parallel }r{\mathrm{\parallel }}_{\rho }$ for small $\rho$ by step 2.1; decreasing $\rho$ makes this norm less than $1\mathrm{/}2$. Thus $q=(1\mathrm{/}{p}_0)\mathrm{\#}{\sum }_{\mathrm{\ell }\ge 0}{r}^{\mathrm{\#}\mathrm{\ell }}$ has finite norm, because in each degree its defining sum is finite and the geometric norm sum is at most $2$. It follows that $q_j$ have factorial bounds on a fixed smaller neighborhood. This gives a right inverse. The identical construction on the other side gives a left inverse; associativity shows they agree. Truncating at $j<ϵ\mathrm{/}h$ realizes these symbols. A term with index $j$ has modulus at most $C(Chj{)}^j$; for $j\le ϵ\mathrm{/}h$ and $Cϵ<1\mathrm{/}2$ it is at most $C{2}^{-j}$. Two admissible truncations therefore differ by $O(e^{-c\mathrm{/}h})$, also in every fixed derivative by [F1].

The actual composition and inverse statements invoked here are the external results [F9]–[F10]. The following calculation identifies their specialization and symbol; it is not an independent reconstruction of that analytic contour calculus. Products of these kernels are actual iterated contour integrals. All weights needed here are of this form, with $S$ the packet phase and its holomorphic critical value; thus the saddle phases used for contour comparison are holomorphic. Their composed contour is good: if the inner integration has loss $c\mathrm{\mid }y-y(z){\mathrm{\mid }}^2$ and the outer one loss $c\mathrm{\mid }z-z(x){\mathrm{\mid }}^2$, their sum controls the square distance from the combined critical point, since the critical-point maps have bounded derivatives. The holomorphic Morse coordinates [F2] identify this contour and the contour obtained by doing the integrations in the other order with graphs over a common negative-definite real subspace. Such graphs have slope less than one after a linear scaling; their straight interpolation has the same positive quadratic margin on a smaller ball. Stokes therefore changes the order and reduces the composition, with exponentially small boundary errors. Taylor expansion in $y-x$, followed by integration by parts in $\theta$, gives the full symbol $p_{\mathrm{r}\mathrm{e}\mathrm{d}}(x,\xi ;h)\sim {\sum }_{\gamma }\frac{h^{\mathrm{\mid }\gamma \mathrm{\mid }}}{i^{\mathrm{\mid }\gamma \mathrm{\mid }}\gamma !}{\mathrm{\partial }}_{\xi }^{\gamma }{\mathrm{\partial }}_y^{\gamma }p(x,y,\xi ;h){\mathrm{\mid }}_{y=x}\mathrm{.}$ For a product this is exactly the $\mathrm{\#}$ formula in step 1.1. To estimate rather than just manipulate this expansion, apply [F3] on the good contour after its quadratic normalization. Cauchy bounds for the analytic amplitudes give $C^{N+1}N!h^N$ for the remainder after total degree $N$; each of the finitely many symbol indices contributes the convolution estimates in step 2.1. Taking $N=\lfloor ϵ\mathrm{/}h\rfloor$ makes this $O(e^{-c\mathrm{/}h})$. The actual operator consequences are supplied by [F9]–[F10]: a zero reduced symbol acts by an exponentially small operator, and the inverse symbol of step 3.1 gives an actual two-sided inverse modulo such errors. Fixed-contour local remainders and symbols are holomorphic in $x$ by the fixed-offset representation in step 1.2, so Cauchy on a further smaller neighborhood controls their fixed derivatives. Errors involving smooth spatial cutoffs or moving outer boundaries are instead differentiated by the chain rule under the fixed-parameter boundary integrals. Each of their fixed $m$ derivatives costs at most $C_m{h}^{-M_m}$, or an additional fixed power of $N$ for a finite-order localizer with $N$ proportional to $h^{-1}$. The boundary has a fixed positive phase margin, so these polynomial losses are absorbed by reducing that margin. No Cauchy estimate is applied to a merely smooth cutoff error.

The same calculation permits a nonstandard holomorphic phase $S(x,y,\theta )$ with $S(x,x,\theta )=0$ and invertible mixed derivative $S_{x\theta }$ at the diagonal. Indeed $S(x,y,\theta )=(x-y)\cdot \mathrm{\Theta }(x,y,\theta ),\mathrm{\Theta }={\int }_0^1{S}_x(y+t(x-y),y,\theta )dt\mathrm{.}$ By [F5], $\theta \mapsto \mathrm{\Theta }$ is a local biholomorphism. The change of variable, including its holomorphic Jacobian, gives the standard kernel of step 1.2. A good contour for one kernel is good for the other because the exponent is identical. Ellipticity is preserved by the nonzero Jacobian. Hence the composition and inverse theorems [F9], with the symbol identification above, hold for these phases too.

For the given positive phase form the kernel $A(x,y;h)=h^{-3n\mathrm{/}2}\int e^{i(\varphi (x,q,p)-{\varphi }^{\ast }(y,q,p))\mathrm{/}h}a(x,q,p;h)dqdp\mathrm{.}$ Locally this is a kernel of the type just constructed. To see this, first integrate in $q$ by analytic stationary phase. At $x=y=q$ the $q$ derivative vanishes, and the imaginary part of its Hessian is positive definite because $\mathrm{Im}\varphi (x,q,p)+\mathrm{Im}\varphi (y,q,p)\ge c(\mathrm{\mid }x-q{\mathrm{\mid }}^2+\mathrm{\mid }y-q{\mathrm{\mid }}^2)$. Its critical value $S(x,y,p)$ is zero at $x=y$, and $S_x(x,x,p)=-p$. Thus $S_{xp}=-I$ there. The resulting kernel has prefactor $h^{-n}$ and a classical analytic amplitude whose leading term is a nonzero Gaussian determinant times $a_0(x,x,p)$. It is elliptic. The actual elliptic inverse theorem [F9], with the phase and symbol identified in steps 1.2, 3.1, 4.1 and 5.1, gives an analytic operator $B$ with $AB=BA=I$ modulo exponential errors on the appropriate holomorphic germs.

Real realization, nested-domain composition, and all fixed-derivative localization bounds are the external input [F10]. We identify its hypotheses here, preserving the local contour and cutoff calculations. Restrict $(q,p)$ to $W$ and insert a fixed smooth spatial cutoff equal to one near $x=y=q$. Away from this diagonal, positivity supplies $\mathrm{Im}(\varphi (x,\alpha )-{\varphi }^{\ast }(y,\alpha ))\ge d>0$ on the cutoff transition. When the kernel acts on a holomorphic packet $v_{\beta }(y)=t(y,\beta ;h)e^{i\psi (y,\beta )\mathrm{/}h}$, the total phase has its unique nearby critical point at $y=q=x$, $p=p_{\beta }$ when $x=q_{\beta }$. In a small neighborhood it satisfies the stronger contour estimate $\mathrm{Im}{S}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\ge c(\mathrm{\mid }x-q{\mathrm{\mid }}^2+\mathrm{\mid }y-q{\mathrm{\mid }}^2+\mathrm{\mid }p-p_{\beta }{\mathrm{\mid }}^2)$ after deforming $y$ by $y\mapsto y+i\delta \overline{{\mathrm{\partial }}_y{S}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}$ near the diagonal. Here Taylor's formula gives an added $\delta \mathrm{\mid }{\mathrm{\partial }}_y{S}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}{\mathrm{\mid }}^2+O({\delta }^2\mathrm{\mid }{\mathrm{\partial }}_y{S}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}{\mathrm{\mid }}^2)$; ${\mathrm{\partial }}_y{S}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=p-p_{\beta }+O(\mathrm{\mid }x-q\mathrm{\mid }+\mathrm{\mid }y-q\mathrm{\mid })$, so sufficiently small $\delta$ proves the estimate. For complex $x$ nearby continue the critical point by [F5] and use the quadratic coordinates to deform a further $O(\mathrm{\mid }x-q_{\beta }\mathrm{\mid })$, leaving the outer boundary fixed. The good-contour estimates of step 4.1 then apply uniformly. On the spatial cutoff transition the original positive margin persists; on $\mathrm{\partial }W$, for $\beta \in W_0$, the displayed estimate supplies a fixed margin. Every swept boundary integral is consequently $O(e^{-c\mathrm{/}h})$. For a fixed number of derivatives, its smoothly parameterized compact boundary, cutoff and phase contribute only fixed powers of $h^{-1}$ and, for finite-order localizers, of $N$. Those powers are absorbed into a smaller exponential margin. Holomorphic fixed-domain contour remainders separately have the Cauchy derivative bounds of step 4.1. If finite-order localization is needed in an integration by parts, [F4] with $N=\lfloor ϵ\mathrm{/}h\rfloor$ gives a factor at most $(ChN{)}^N$ on the nonstationary part; choose $Cϵ<1\mathrm{/}2$. This verifies the real realization of the compositions and of the identity on each such packet.

The packet-action lemma, Theorem 5.3 and Proposition 6.2 in [F10] supply the actual realization of this step. Applying $B$ to $v_{\beta }$ gives $e^{i\psi (y,\beta )\mathrm{/}h}b(y,\beta ;h)$ modulo exponential errors, where $b$ is a classical analytic amplitude bounded by a fixed power of $h^{-1}$. This follows from the same composition calculation, treating $\beta$ as a parameter and taking the weight $\rho =-\mathrm{Im}\psi$ in step 1.2; the combined critical point is the diagonal, and steps 4.1 and 7.1 give the factorial remainder on smaller fixed neighborhoods. Therefore $AB{v}_{\beta }=v_{\beta }+R_{\beta }$, and its realized real-variable expression is the representation in the statement with $K(\beta ,\alpha ;h)=h^{-3n\mathrm{/}2}\int e^{i(\psi (y,\beta )-{\varphi }^{\ast }(y,\alpha ))\mathrm{/}h}b(y,\beta ;h)dy\mathrm{.}$ The imaginary part of this exponent is at least $c(\mathrm{\mid }y-q_{\beta }{\mathrm{\mid }}^2+\mathrm{\mid }y-q_{\alpha }{\mathrm{\mid }}^2)$ for real parameters, so the compact integral is bounded by a power of $h^{-1}$, uniformly in $\alpha \in W$, $\beta \in W_0$. Errors and cutoff transitions retain the derivative bounds of step 7.1. Pairing the representation with $u$ is legitimate by [F7]; its finite-order seminorm bound makes $⟨u,R_{\beta }⟩$ exponential. The integral of a uniformly exponentially small source transform times the polynomially bounded $K$ is exponential too. Interchange the two elliptic packet families to obtain the reverse implication.

Finally use the reference phase ${\varphi }_G(y,q,p)=(q-y)\cdot p+\frac{i\mathrm{\mid }p\mathrm{\mid }}{2}\mathrm{\mid }q-y{\mathrm{\mid }}^2\mathrm{.}$ It is holomorphic on a complex neighborhood of any nonzero real $p_0$, choosing the branch of $\sqrt{p\cdot p}$ positive there. Its packet with parameter $\lambda$ is exactly the unit-direction Gaussian packet with frequency parameter $\lambda \mathrm{\mid }p\mathrm{\mid }$ and direction $p\mathrm{/}\mathrm{\mid }p\mathrm{\mid }$. On a fixed small parameter neighborhood $\mathrm{\mid }p\mathrm{\mid }$ is bounded above and below by positive constants, so the exponential criteria agree; bounded frequencies are absorbed into the constant. All amplitude and inverse constructions above involve finite sums or a uniquely determined formal recursion and its finite truncation. No choice axiom is used.