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The linear homological equation $ν \cdot \partial_{ϑ} u = v$ on a Diophantine label set, with uniform parameter bounds and conservative loss

Statement

Let $n \geq 1$ , $τ > n - 1$ , $α > 0$ , and let $Ω_{α, τ}$ be the label set of Analytic KAM data: angular tori, action-angle neighbourhoods, Fourier cutoffs and Diophantine frequency labels. Let $r, s > 0$ , $K \geq 0$ , and suppose one of the following periodic, zero-mean data conditions holds.

(G) General case: for each real $ν \in Ω_{α, τ}$ , $v (\cdot, \cdot, ν)$ is holomorphic on $B_{r} \times S_{s}$ , with $N : = \sup_{ν \in Ω_{α, τ}} \sup_{I \in B_{r}, ϑ \in S_{s}} ∣ v (I, ϑ, ν) ∣ < \infty .$ This is a family of phase-holomorphic functions; no holomorphy on a complex parameter neighbourhood is assumed.
(T) Truncated case: $h > 0$ , and $v$ is jointly holomorphic on $D_{r, s, h} (Ω_{α, τ})$ , with $N : = ∣ v ∣_{r, s, h} < \infty$ and $v_{k} = 0$ for $∣ k ∣_{1} > K$ . If $K > 0$ , assume $h \leq α / (2 K^{τ + 1})$ . If $K = 0$ , the zero-mean datum is identically zero and no divisor restriction on $h$ is needed.

Put $α_{0} : = α$ in (G) and $α_{0} : = α / 2$ in (T). In (G), for a family $w$ of phase functions write $∥ w ∥_{a, b}^{G} : = \sup_{ν \in Ω_{α, τ}} \sup_{I \in B_{a}, ϑ \in S_{b}} ∣ w (I, ϑ, ν) ∣;$ this is explicitly a supremum of phase-space suprema over real labels. In (T), norms retain the joint convention $∣ \cdot ∣_{a, b, h}$ .

For every parameter of the relevant case and every nonzero $k$ in the Fourier support of $v$ , $∣ k \cdot ν ∣ \geq α_{0} ∣ k ∣_{1}^{- τ}$ . For nonzero $k$ , define $a_{k} : = v_{k} / (i k \cdot ν)$ in (G), and for $0 < ∣ k ∣_{1} \leq K$ in (T); set $a_{k} : = 0$ for $∣ k ∣_{1} > K$ in (T). The series $u (I, ϑ, ν) : = \sum_{k \in Z^{n}, k \neq 0} a_{k} (I, ν) e^{i k \cdot ϑ}$ converges absolutely and uniformly on $B_{r} \times S_{s - 2 σ}$ at every fixed label in (G), uniformly over those real labels, and on $D_{r, s - 2 σ, h} (Ω_{α, τ})$ in (T), whenever $0 < σ \leq 1$ and $2 σ < s$ . Its sum is phase-holomorphic at each label in (G), jointly holomorphic in (T), periodic, zero-mean, and satisfies $ν \cdot \partial_{ϑ} u = v$ . Any two periodic holomorphic zero-mean solutions on a common such case domain agree there.

For every $0 < σ \leq 1$ with $4 σ < s$ , every $i, j \in {1, \dots, n}$ , and every $0 < δ < r$ , the following bounds hold in the joint case (T): $∣ u ∣_{r, s - 2 σ, h} \leq \frac{C_{n, τ}}{α_{0}} σ^{- (τ + n)} N, ∣ \partial_{ϑ_{j}} u ∣_{r, s - 3 σ, h} \leq \frac{C_{n, τ}}{α_{0}} σ^{- (τ + n + 1)} N,$ $∣ \partial_{I_{j}} u ∣_{r - δ, s - 2 σ, h} \leq \frac{C_{n, τ}}{α_{0} δ} σ^{- (τ + n)} N, ∣ \partial_{ϑ_{i}} \partial_{ϑ_{j}} u ∣_{r, s - 4 σ, h} \leq \frac{C_{n, τ}}{α_{0}} σ^{- (τ + n + 2)} N,$ $∣ \partial_{ϑ_{i}} \partial_{I_{j}} u ∣_{r - δ, s - 3 σ, h} \leq \frac{C_{n, τ}}{α_{0} δ} σ^{- (τ + n + 1)} N, ∣ \partial_{I_{i}} \partial_{I_{j}} u ∣_{r - δ, s - 2 σ, h} \leq \frac{C_{n, τ}}{α_{0} δ^{2}} σ^{- (τ + n)} N .$ In (G), the same six inequalities hold with each joint norm $∣ w ∣_{a, b, h}$ replaced by the explicit supremum $∥ w ∥_{a, b}^{G}$ ; thus no output bound is a norm on the empty set $O_{0} (Ω_{α, τ})$ .

In (T), $u_{k} = 0$ for $∣ k ∣_{1} > K$ ; in particular $u$ and its displayed derivatives have the same Fourier support as $v$ .

Facts & Assumptions

Given: The dimension $n$ , exponent $τ > n - 1$ , constant $α > 0$ , label set $Ω_{α, τ}$ , widths $r, s$ and (in (T)) $h > 0$ , cutoff $K$ , and the periodic zero-mean data of case (G) or (T), with the bound $N$ defined in the corresponding case above.

[F1]

The definition of $D_{r, s, h}$ , holomorphy, Fourier coefficients, angular mean, Fourier cutoff, and the $(α, τ)$ -Diophantine condition $∣ k \cdot ν ∣ \geq α ∣ k ∣_{1}^{- τ}$ for every nonzero $k$ (Analytic KAM data: angular tori, action-angle neighbourhoods, Fourier cutoffs and Diophantine frequency labels).

[F2]

In (G), for each fixed label the phase Fourier series of $v$ equals $v$ on each smaller strip, its coefficients obey $∣ v_{k} ∣ \leq N e^{- (s - ρ) ∣ k ∣_{1}}$ for $0 < ρ < s$ , and it can be differentiated termwise locally uniformly. In (T), the same assertions hold jointly on smaller action, angle and parameter domains (Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods clauses 1 and 2).

[F3]

The lattice sums include $\sum_{k \neq 0} ∣ k ∣_{1}^{τ} e^{- σ ∣ k ∣_{1}} \leq C_{n, τ} σ^{- (τ + n)}$ for $0 < σ \leq 1$ , and the corresponding sums with one or two extra powers of $∣ k ∣_{1}$ (Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods clause 3).

[F4]

Locally uniformly convergent holomorphic series may be differentiated termwise, and their sums are holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly, A locally uniformly convergent series of holomorphic functions may be differentiated term by term, A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc).

Proof

Given: In (T), every quotient is formed only for retained modes $0 < ∣ k ∣_{1} \leq K$ ; discarded coefficients are zero. In (G), each fixed parameter is a real Diophantine label and all nonzero modes are retained.

Proof technique: direct; estimate the divisors, sum the phase Fourier series, and use coordinate Cauchy estimates for its action derivatives.

1.1givenF1algebra

Small divisors. In (G), $∣ k \cdot ν ∣ \geq α ∣ k ∣_{1}^{- τ}$ for each [given, F1, algebra] real label and every nonzero mode. In (T), fix $ν \in O_{h} (Ω_{α, τ})$ and a retained mode $0 < ∣ k ∣_{1} \leq K$ . There is a real label $ν_{0}$ with $∣ ν - ν_{0} ∣_{\infty} < h$ , so $∣ k \cdot ν ∣ \geq α ∣ k ∣_{1}^{- τ} - ∣ k ∣_{1} h \geq (α / 2) ∣ k ∣_{1}^{- τ},$ since $h ∣ k ∣_{1}^{τ + 1} \leq h K^{τ + 1} \leq α / 2$ . No discarded mode is divided by its divisor.

2.1step 1.1F2F3F4algebra

Convergence and holomorphy. Let $0 < σ \leq 1$ with $2 σ < s$ . For [step 1.1, F2, F3, F4, algebra] each label in (G), and uniformly over those labels by the definition of $N$ , or on the joint domain in (T), [F2] and step 1.1 give $\sum_{k \neq 0} ∣ \frac{v_{k}}{i k \cdot ν} e^{i k \cdot ϑ} ∣ \leq \frac{N}{α_{0}} \sum_{k \neq 0} ∣ k ∣_{1}^{τ} e^{- σ ∣ k ∣_{1}} \leq \frac{C_{n, τ}}{α_{0}} σ^{- (τ + n)} N$ by [F3]. Thus the series is absolutely and uniformly convergent on the specified phase strip (jointly in all variables in (T)); it is holomorphic in the phase variables in (G) and jointly holomorphic in (T), by [F4]. Each term has zero angular mean, hence $[u] = 0$ . Taking the appropriate supremum gives the first bound in clause 2, using $∥ \cdot ∥^{G}$ in (G).

3.1step 2.1F2F3F4algebra

The equation and angular derivatives. On every strictly smaller strip, [step 2.1, F2, F3, F4, algebra] the differentiated series is locally uniformly convergent by [F2] and [F3]. Termwise differentiation yields $\partial_{ϑ_{j}} u = \sum_{k \neq 0} \frac{v_{k} k_{j}}{k \cdot ν} e^{i k \cdot ϑ}, ν \cdot \partial_{ϑ} u = \sum_{k \neq 0} v_{k} e^{i k \cdot ϑ} = v - [v] = v,$ where the last equality follows from [F2]. The extra factor $∣ k_{j} ∣ \leq ∣ k ∣_{1}$ and the two angle margins in clause 2 give the stated one- and two-angle bounds using [F3].

3.2step 1.1step 2.1F2F3algebra

Action derivatives. By [F2], $v_{k}$ is holomorphic in the action on $B_{r}$ [step 1.1, step 2.1, F2, F3, algebra] and $∣ v_{k} ∣ \leq N e^{- (s - ρ) ∣ k ∣_{1}}$ . The coordinate Cauchy estimates of Fourier decay, Cauchy derivative bounds, lattice shell counts and the exponential small-divisor sums on action-angle neighbourhoods clause 2, with action margin $δ$ , give on $B_{r - δ}$ $∣ \partial_{I_{j}} v_{k} ∣ \leq \frac{N}{δ} e^{- (s - ρ) ∣ k ∣_{1}}, ∣ \partial_{I_{i}} \partial_{I_{j}} v_{k} ∣ \leq \frac{2 N}{δ^{2}} e^{- (s - ρ) ∣ k ∣_{1}} .$ These estimates are pointwise at every label in (G), uniformly by $N$ , and jointly on the smaller parameter domain in (T). Division by the divisor and summation using [F3] prove the action and mixed action-angle bounds in clause 2, with the explicit label supremum in (G). The locally uniform convergence also justifies termwise differentiation.

3.3givenstep 2.1algebra

Truncation. In (T), $v_{k} = 0$ for $∣ k ∣_{1} > K$ and the solution coefficient [given, step 2.1, algebra] is defined to be zero there, so every displayed derivative has support in $∣ k ∣_{1} \leq K$ .

4.1step 1.1step 2.1step 3.1F2algebra∎

Uniqueness. For a fixed real label in (G), Fourier coefficients of any [step 1.1, step 2.1, step 3.1, F2, algebra] zero-mean phase-holomorphic solution $w$ obey $(i k \cdot ν) w_{k} = v_{k}$ for every nonzero mode, so every coefficient agrees with $u$ . In (T), the same comparison applies on the open complex parameter domain. For retained modes, equality follows from step 1.1. For a discarded mode, $v_{k} = 0$ and $(k \cdot ν) w_{k} = 0$ ; away from the hyperplane $k \cdot ν = 0$ , $w_{k} = 0$ , and continuity extends this across that hyperplane because its complement is dense in each open ball. The zero coefficient vanishes by the zero-mean condition. Fourier uniqueness from [F2] gives equality on the common domain in either case.

Remarks

What "truncated" refers to. Case (T) is the situation in the KAM step: the unknown is solved only for the Fourier-truncated data, and the parameter margin $h$ has to be small enough that the Diophantine bound survives on the complex parameter neighbourhood, at the price of replacing $α$ by $α / 2$ . No loss in $α$ occurs in the parameter-free case (G).
Conservatism. The exponents are exactly those proved by summing the lattice shells; the number $τ + n + 2$ appearing in the second derivatives is the largest loss used downstream, and no sharper small-divisor theorem is invoked.

The linear homological equation $ν \cdot \partial_{ϑ} u = v$ on a Diophantine label set, with uniform parameter bounds and conservative loss

Statement

Facts & Assumptions

Proof

Remarks

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The linear homological equation ν⋅∂ϑu=vν⋅∂ϑ​u=v on a Diophantine label set, with uniform parameter bounds and conservative loss

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Facts & Assumptions

Proof

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The linear homological equation $ν \cdot \partial_{ϑ} u = v$ on a Diophantine label set, with uniform parameter bounds and conservative loss