Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods

Fix all variables except one real angle ${\vartheta }_j$ and put $f_j(z)=g(I,{\vartheta }_1,\dots ,z,\dots ,{\vartheta }_n,\nu )e^{-i{k}_{j}z}$, with the absent parameter omitted in the phase-only case. Since $k_j\in \mathbb{Z}$ and $g$ is $2\pi$-periodic, $f_j$ is periodic and holomorphic for $\mathrm{\mid }\mathrm{Im}z\mathrm{\mid }<s$. For $0<\delta <s$, shift the integration segment $[0,2\pi ]$ to $[0,2\pi ]-i\delta \mathrm{sgn}(k_j)$; the rectangle integral is zero by [F2], and the two vertical integrals cancel by periodicity. On the shifted segment $\mathrm{\mid }{e}^{-i{k}_{j}z}\mathrm{\mid }=e^{-\delta \mathrm{\mid }{k}_j\mathrm{\mid }}$, so the $j$th Fourier integral gains that factor. Applying this in each coordinate, using [F6], gives $\mathrm{\mid }{g}_k(I,\nu )\mathrm{\mid }\le M{e}^{-\delta \mathrm{\mid }k{\mathrm{\mid }}_1}$. For $0<\sigma <s$ take $\delta =s-\sigma$; for $\sigma =s$ the claimed bound is the trivial estimate $\mathrm{\mid }{g}_k\mathrm{\mid }\le M$.

The number of signed integer vectors with $\mathrm{\mid }k{\mathrm{\mid }}_1=m$ is at most $2^n$ times the number of nonnegative integer $n$-tuples with sum $m$. Choosing the first $n-1$ entries, each between $0$ and $m$, gives at most $2^n(m+1{)}^{n-1}\le 2^{2n-1}{m}^{n-1}\le 4^n{m}^{n-1}$ vectors. Therefore ${\sum }_k{e}^{-\sigma \mathrm{\mid }k{\mathrm{\mid }}_1}=({\sum }_{q\in \mathbb{Z}}{e}^{-\sigma \mathrm{\mid }q\mathrm{\mid }}{)}^n\le C_n{\sigma }^{-n}\mathrm{.}$ For $p=\tau +n-1\ge 0$, comparison on $x\in [m-1,m]$ gives $m^p{e}^{-\sigma m}\le {\int }_{m-1}^m(x+1{)}^p{e}^{-\sigma x}dx$. Thus ${\sum }_{m\ge 1}{m}^p{e}^{-\sigma m}\le {\sigma }^{-(p+1)}{\int }_0^{\mathrm{\infty }}(y+1{)}^p{e}^{-y}dy,$ which is finite and proves the positive-power bound after applying the shell count. If $\tau \ge n-1$, then $m^{n-1-\tau }\le 1$ and ${\sum }_{m\ge 1}{e}^{-\sigma m}=1\mathrm{/}(e^{\sigma }-1)\le 1\mathrm{/}\sigma$; this proves the negative-power bound.

Products of local power series are local power series and satisfy the pointwise product bound. For a uniform limit on the full domain, continuity passes to the limit and every coordinate slice is a locally uniform limit of one-variable holomorphic functions; [F4] makes the limit separately holomorphic and hence holomorphic, while passing to the limit in the pointwise bound gives norm at most $M$.

For $0<\sigma \le 1$ and $2\sigma <s$, the estimate of step 1.1 gives on $S_{s-2\sigma }$ $\mathrm{\mid }{g}_k(I,\nu )e^{ik\cdot \vartheta }\mathrm{\mid }\le M{e}^{-\sigma \mathrm{\mid }k{\mathrm{\mid }}_1}\mathrm{.}$ The product of one-dimensional geometric sums in [F8] is at most $C_n{\sigma }^{-n}$; hence the Fourier series is absolutely and uniformly convergent on the full stated open product. Its sum is continuous and, by locally uniform convergence and [F4], holomorphic. For fixed non-angle variables, its Fourier coefficients are exactly $g_k$, by uniform convergence and [F6]. The difference from $g$ therefore has zero Fourier coefficients and vanishes on the real torus by [F5], applied successively in the $n$ angle coordinates. At each real angle point all real derivatives of this difference vanish; holomorphicity identifies these with its complex Taylor coefficients, so its local power series is zero on a complex polydisc. The strip $S_{s-2\sigma }$ is connected, so [F7] gives equality on the full complex strip. For joint-parameter data, the Fourier coefficient is holomorphic in $(I,\nu )$: its Riemann sums are holomorphic and converge uniformly on every compact subset of $B_r\times O_h(\mathrm{\Omega })$, so [F4] and separate holomorphy apply.

The coordinate Cauchy estimate [F3] on a radius-$\delta$ disc about $I_j$ gives the first action-derivative bound, and the same argument on the angle and parameter coordinates gives the other first-derivative bounds. For a second derivative in the same coordinate, [F3] with order $2$ gives the factor $2$; for distinct coordinates, use the iterated Cauchy formula on the product of their coordinate circles, obtaining $M\mathrm{/}({\delta }_i{\delta }_j)$. This yields the stated action, angle, and mixed estimates on their strict smaller domains.

Let $K\ge 0$ and choose the coefficient-shift margin $\delta =s-\sigma \mathrm{/}2$, which lies strictly between $0$ and $s$. On $S_{s-2\sigma }$ each discarded Fourier term is bounded by $M{e}^{-3\sigma \mathrm{\mid }k{\mathrm{\mid }}_1\mathrm{/}2}$. For $\mathrm{\mid }k{\mathrm{\mid }}_1>K$, $e^{-3\sigma \mathrm{\mid }k{\mathrm{\mid }}_1\mathrm{/}2}\le e^{-\sigma K}{e}^{-\sigma \mathrm{\mid }k{\mathrm{\mid }}_1\mathrm{/}2}$; summing with [F8] and step 1.2 gives the stated truncation bound, after absorbing $2^n$ into $C_n$.

On any compact subset of the smaller open product in clause 1 there are positive action, angle, and (in the joint case) parameter margins to the boundary of the original domain. Apply step 1.1 with a fixed strictly smaller strip width, then apply the coordinate Cauchy estimates of step 2.2 on those margins. Each differentiated Fourier term is bounded by a polynomial in $\mathrm{\mid }k{\mathrm{\mid }}_1$ times $M{e}^{-\delta \mathrm{\mid }k{\mathrm{\mid }}_1}$ for some $\delta >0$ depending on the compact set. The shell comparison of step 1.2 makes these bounds summable. The Weierstrass test therefore gives locally uniform convergence of each termwise-differentiated series, and [F4] identifies those sums with the corresponding derivatives.