Affine Taylor splitting in the actions and conservative Fourier truncation of a small perturbation

Structure of $Q$ and $R$. $Q$ is defined from the holomorphic functions $P(0,\cdot ,\cdot )$ and ${\mathrm{\partial }}_{I_j}P(0,\cdot ,\cdot )$, which are holomorphic on $B_r\times S_s\times O_h$; hence $Q$ is holomorphic there and affine in $I$. Its Fourier coefficients in $\vartheta$ are those of $P(0,\cdot )$ and ${\mathrm{\partial }}_{I_j}P(0,\cdot )$; the cutoff $R=T_{K}Q$ is the corresponding finite trigonometric polynomial, again holomorphic and affine in $I$, with $R_0=T_K(P(0,\cdot ))$ and $R_j$ the truncated coefficients, so $[R]=[Q]$ and $R-[R]$ is zero-mean. This is clause 2.

Bounds for $Q$. By [F2] with radii increasing to $r$, at the centre $I=0$ one has $\mathrm{\mid }P(0,\vartheta ,\nu )\mathrm{\mid }\le \epsilon$ and $\mathrm{\mid }{\mathrm{\partial }}_{I_j}P(0,\vartheta ,\nu )\mathrm{\mid }\le \epsilon \mathrm{/}r$; hence on $\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}<r$ $\mathrm{\mid }Q(I,\vartheta ,\nu )\mathrm{\mid }\le \epsilon +{\sum }_{j=1}^n\frac{\epsilon }{r}\mathrm{\mid }{I}_j\mathrm{\mid }\le (n+1)\epsilon \mathrm{.}$

Taylor remainder. Fix $(I,\vartheta ,\nu )$ with $\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}<2\eta r$. By [F3], $P(I)-Q(I)={\int }_0^1(D_{I}P(tI)-D_{I}P(0))\cdot Idt$; for $\mathrm{\mid }J{\mathrm{\mid }}_{\mathrm{\infty }}\le 2\eta r\le r\mathrm{/}2$ [F2] with $\delta =r\mathrm{/}2$ gives $\mathrm{\mid }{D}_I^{2}P(J)\mathrm{\mid }\le 2\epsilon \mathrm{/}(r\mathrm{/}2{)}^2=8\epsilon \mathrm{/}{r}^2$ (maximum over the coordinate second derivatives), so for every $t\in [0,1]$, $\mathrm{\mid }{D}_{I}P(tI)-D_{I}P(0)\mathrm{\mid }\le n\mathrm{\mid }tI{\mathrm{\mid }}_{\mathrm{\infty }}{\sup }_{\mathrm{\mid }J{\mathrm{\mid }}_{\mathrm{\infty }}\le 2\eta r}\mathrm{\mid }{D}_I^{2}P(J)\mathrm{\mid }\le 2n\eta r\cdot \frac{8\epsilon }{r^2}=\frac{16n\eta \epsilon }{r}\mathrm{.}$ Therefore $\mathrm{\mid }P-Q\mathrm{\mid }\le {\sup }_t\mathrm{\mid }{D}_{I}P(tI)-D_{I}P(0){\mathrm{\mid }}_1\mathrm{\mid }I{\mathrm{\mid }}_{\mathrm{\infty }}\le n\cdot 16n\eta \epsilon \mathrm{/}r\cdot 2\eta r=32n^2{\eta }^2\epsilon$, which is clause 1 with $C=32n^2$.

Size bounds for $R$. By step 1.2 and [F1], each coefficient $q_k$ of $Q$ satisfies $\mathrm{\mid }{q}_k(I,\nu )\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}_{r,s,h}{e}^{-(s-\sigma )\mathrm{\mid }k{\mathrm{\mid }}_1}$ (the decay of Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods clause 1 with the margin $\sigma \le s$). Hence, on $B_r\times S_{s-2\sigma }\times O_h$, $\mathrm{\mid }R\mathrm{\mid }\le {\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_1\le K}\mathrm{\mid }{q}_k\mathrm{\mid }{e}^{(s-2\sigma )\mathrm{\mid }k{\mathrm{\mid }}_1}\le \mathrm{\mid }Q{\mathrm{\mid }}_{r,s,h}{\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_1\le K}{e}^{-\sigma \mathrm{\mid }k{\mathrm{\mid }}_1}\le (n+1)\epsilon C_n{\sigma }^{-n},$ by clause 3 of Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods, which is the first bound of clause 3. For $R_j$: $R_j$ is the cutoff of ${\mathrm{\partial }}_{I_j}P(0,\cdot ,\cdot )$, whose coefficients are bounded by $(\epsilon \mathrm{/}r)e^{-(s-\sigma )\mathrm{\mid }k{\mathrm{\mid }}_1}$ by [F2] and the same decay; the identical summation gives ${\max }_j\mathrm{\mid }{R}_j{\mathrm{\mid }}_{r,s-2\sigma ,h}\le C_n{\sigma }^{-n}\epsilon \mathrm{/}r$. For ${\mathrm{\partial }}_{{\vartheta }_i}R$: by [F2] with ${\delta }^{\mathrm{\prime }}=\sigma$ and $\delta =r\mathrm{/}2$, $\mathrm{\mid }{\mathrm{\partial }}_{{\vartheta }_i}P(0,\cdot ,\cdot )\mathrm{\mid }\le \epsilon \mathrm{/}\sigma$ and $\mathrm{\mid }{\mathrm{\partial }}_{{\vartheta }_i}{\mathrm{\partial }}_{I_j}P(0,\cdot ,\cdot )\mathrm{\mid }\le 4\epsilon \mathrm{/}(r\sigma )$; hence $\mathrm{\mid }{\mathrm{\partial }}_{{\vartheta }_i}Q\mathrm{\mid }\le \epsilon \mathrm{/}\sigma +n\cdot 4\epsilon \mathrm{/}(r\sigma )\cdot r\le (1+4n)\epsilon \mathrm{/}\sigma$ on $B_r\times S_{s-\sigma }\times O_h$, and the cutoff of this (holomorphic, affine-in-$I$) function on the smaller strip is bounded by $C_n{\sigma }^{-n}\cdot (1+4n)\epsilon \mathrm{/}\sigma$, which is the third bound of clause 3.

Truncation error. Applying [F2]'s truncation estimate to $Q$ with $\tau =\sigma \le \min (s\mathrm{/}2,1)$ and the bound $\mathrm{\mid }Q{\mathrm{\mid }}_{r,s,h}\le (n+1)\epsilon$ of step 1.2 gives $\mathrm{\mid }Q-R{\mathrm{\mid }}_{r,s-2\sigma ,h}\le C_n{\sigma }^{-n}{e}^{-\sigma K}(n+1)\epsilon$, which is clause 4 after enlarging the constant; note $e^{-\sigma K}\le e^{-\sigma K\mathrm{/}2}$, so the displayed form with $e^{-\sigma K\mathrm{/}2}$ holds as well.