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Long time existence for the semi-linear beam equation
on irrational tori of dimension two
Rafik Imekraz
To cite this version:
Long time existence for the semi-linear
beam equation on irrational tori of dimension two
Rafik Imekraz
Institut de Math´ematiques de Bordeaux, UMR 5251 du CNRS, Universit´e de Bordeaux, 351, cours de la Lib´eration F33405 Talence Cedex, France
May 30, 2016
Abstract
We prove a long time existence result for the semi-linear beam equation with small and smooth initial data. We use a regularizing effect of the structure of beam equations and a very weak separation property of the spectrum of an irrational torus under a Diophantine assumption on the radius. Our approach is inspired from a paper by Zhang about the Klein-Gordon equation with a quadratic potential.
2000 Mathematics Subject Classification : 37K45, 35Q55, 35B34, 35B35 Key words : beam equation, Klein-Gordon equation, normal form, irrational torus
1
Introduction
In this paper we study the solutions of the so-called beam equation with periodic boundary conditions and with a positive and constant potential :
(∂t2+ ∆2+ m2)w = wn+1, (x, y) ∈ [0, 2π] × [0, 2πr], t ∈ R. (1) Here, the length r is a positive number, n ≥ 2 is an integer and m is a positive number. From a Riemannian point of view, the previous setting is equivalent to pose the equation on the compact manifold S1× rS1
so r can be interpreted as a radius. Originally, the beam equation has a physical meaning in dimension 1 because it arises in modeling the oscillations of a uniform beam. In dimension 2, similar equations can be used to model the motion of a clamped plate (see several references in the introduction of [22]). Recent mathematical results have been proven in KAM or scattering frameworks, we refer for instance to the works [15, 16, 21, 23].
Since ∆2 is a fourth-order operator, a natural functional space for the equation (1) is C0((−T, T ), Hs+2
(S1× rS1)) ∩ C1((−T, T ), Hs
(S1× rS1)), T > 0,
s ∈ R. Using the fact that Hs
(S1× rS1) is an algebra for any real number s > 1 and reformulating the beam
equation (1) with the Duhamel formula, one can check that, for any initial data (w(0), ˙w(0)) = (εw0, εw1)
in Hs+2
(S1× rS1) × Hs
(S1× rS1), if ε ∈ (0, 1) is small enough then the beam equation (1) admits a
unique solution w on a time interval of length T & ε−nsuch that
∀t ∈ (−T, T ) ||w(t)||Hs+2(S1×rS1)+ || ˙w(t)||Hs(S1×rS1)≤ Cε.
The time ε−nis called the local existence time. The goal of this paper is to improve the local existence
time ε−n to ε−An for some universal constant A > 1. We will use methods which have been developed
in the framework of the semi-linear Klein-Gordon equation on a compact Riemannian manifold X : (∂2t− ∆ + m2)w = wn+1, x ∈ X,
More precisely, we will adapt the paper [26] to our framework (the differences are explained below). Let us briefly recall what is known about the improvement of the local existence time ε−nfor the equation (2). Using a normal form procedure for a generic parameter m > 0 and for high regularities s 1, one can divide this analysis in two categories :
• if the spectrum of √−∆ is separated. That means that the difference of two successive eigenvalues of √−∆ is uniformly bounded from below. For those manifolds, we can improve the local existence time ε−nto c(A)ε−Anfor any A > 1 and we usually say that an almost global existence holds for the Klein-Gordon equation (2). The simplest manifold we have in mind is the one-dimensional torus T ([6, 1]). The case of the sphere Sd indeed appears as a particular case of Zoll manifolds. Eigenvalues of those manifolds have a property of separation which is weaker than the one above. Nevertheless, Bambusi, Delort, Gr´ebert and Szeftel succeeded in proving the almost global existence for the equation (2) on a Zoll manifold (see the paper [3]). Their proof relies on universal multilinear estimates of eigenfunctions proven by Delort and Szeftel in the paper [10]. We also refer to the papers [17, 2, 4, 9, 19, 8] for more on the subject.
• if the spectrum of √−∆ is not separated. In this category, we have to think to the multi-dimensional torus Td, for instance if d ≥ 4 then the eigenvalues of√−∆ are exactly the numbers√k with k ∈ N (so that we have limk→+∞√k + 1 −√k = 0). The multidimensional torus has been studied by Delort in [7] with a new approach of small divisors and he improved the local existence time from ε−n to ε−n(1+2d) (up to a logarithmic term). This time has been improved to ε−32n by Fang and Zhang
(see [11]) if d is larger than 4. In the last two papers, the harmonic analysis of Td is used, namely the fact that the eigenfunctions of Td
are naturally parametrized by Zd. Later in [26], Zhang proved that
the harmonic analysis is not really necessary to get a long time existence by dealing with the harmonic oscillator (the price to pay is to have a weaker improvement of the local existence time equal to ε−43n).
The author generalized the previous paper for superquadratic oscillators in [20] for which we can reach the time ε−2ndue to a stronger separation of the eigenvalues.
We also refer to the papers [12, 14, 13, 5] for analogue questions on Schr¨odinger equations and references therein.
Therefore, it is a natural question to study the semi-linear Klein-Gordon equation (2) on a compact manifold with very badly separated eigenvalues. Unfortunately, as in lots of other contexts in partial differential equations, it seems that we need a knowledge about the behavior of the eigenvalues. In contrast, working on the beam equation (1) is easier because it admits a slight regularizing effect due to the operator ∆2 of order 4. The irrational torus S1× rS1seems to be a good candidate for at least two
reasons :
i) it is an explicit example of manifold with a very bad spectral behavior (accumulation of eigenval-ues),
ii) if one comes back to the interpretation of S1
×rS1as periodic boundary conditions on [0, 2π]×[0, 2πr],
then one may say that the generic condition is the irrational case r−2∈ Q. In other words, irrational tori happens more often than rational tori.
Roughly speaking, our contribution is an adaptation of the method of [26] to get a long time existence result for (1) on an irrational torus S1× rS1, we use the property that the very bad separation of the
spectrum is counterbalanced by a natural regularizing effect of the beam equation.
From now, we go into details of the description of our main result (Theorem 1.2 below) and we explain the differences with previous works. The first thing to do is looking at the spectrum of the Laplace-Beltrami operator of the torus X = S1× rS1:
Sp(−∆) := {p2+ r−2q2, (p, q) ∈ Z2}.
multiplicities of eigenvalues are equal to 1, 2 or 4. Let us sort the spectrum of√1 − ∆ as an increasing and positive sequence (λk)k≥1 without counting multiplicities :
Sp(I − ∆) = {λ21< λ 2 2< λ
3
3< . . . }.
From the Weyl law or by a direct computation, one checks that λkis asymptotically equivalent to √2rπ
√ k as k tends to infinity. Such an asymptotic equivalent is simple but not sufficient to write a normal form procedure. We indeed need to understand the behavior of differences of two eigenvalues, or equivalently to get suitable approximations of the number r−2 by the rational numbers. As often in dynamical systems, Diophantine conditions naturally appear.
Definition 1.1. An irrational real number R is Diophantine (and we write R ∈ D) if the following holds
∀µ > 2 ∃C(R, µ) > 0 ∀P Q∈ Q R −P Q ≥ C(R, µ) |Q|µ . (3)
The condition µ > 2 in (3) will play a role in the improvement of the local existence time of the beam equation. Let us recall that in the previous definition, we cannot expect that µ is less than 2 in (3) for an irrational number R > 0 since Dirichlet’s approximation theorem states that there are infinitely many rationals numbers PQ such that
R − P Q < 1 Q2.
The study of irrationality measures of real numbers has a long history and we only state the results which prove that the set D of the Diophantine numbers is not empty :
a) almost any real number R in the sense of Lebesgue belongs to D (this is a classical fact and it is easy to prove by using the convergence of the seriesP Q−(µ−1)
with Q running over N? and µ ∈ (2, +∞)
is fixed),
b) there are also numbers R such that one can choose µ = 2 in (3). Such numbers are called “badly approximable”, for instance irrational quadratic numbers are convenient,
c) any irrational algebraic number belongs to D. For instance, √d
2 belongs to D for any integer d ≥ 2. This is the famous Roth theorem [24].
We refer for instance to [18, Part D] or [25, Chapter II] for more about the theory of Diophantine approximations. We can now state our main result about the beam equation (1).
Theorem 1.2. Assume that r−2 belongs to D and fix A ∈ (1,5
4), there is a zero Lebesgue measure
subset En,r,A⊂ (0, +∞) such that the following holds for any m ∈ (0, +∞)\En,r,A. For any large enough
s 1, for any couple of real-valued functions (w0, w1) ∈ Hs+2(S1× rS1) × Hs(S1× rS1) with ||w0||Hs+2+
||w1||Hs = 1, there are C, K > 0 such that if ε > 0 is small enough then the beam equation (1) admits a
unique solution
w ∈ C0((−Cε−An, +Cε−An), Hs+2(S1× rS1)) ∩ C1((−Cε−An, +Cε−An), Hs
(S1× rS1)),
with initial data (w(0), ˙w(0)) = (εw0, εw1). Furthermore one has
∀t ∈ (−Cε−An, +Cε−An) ||w(t)||Hs+2+ || ˙w(t)||Hs≤ Kε.
Remark 1.3. For instance, the previous result covers the case of the manifold S1×√3
2S1 thanks to the Roth theorem with with r−2= 1
3
√
4. It would be very interesting to understand if the analogue of Theorem
1.2 for the Klein-Gordon equation (2) is true or not. Although a modification of the condition A ∈ (1,54) may be necessary, the proof would work for the semi-linear equation (∂t2+ |∆|α+ m2)w = wn+1 where |∆|αis a fractional power of the Laplace-Beltrami operator with α > 1. A similar phenomenon is studied
Although the strategy of the proof is quite similar of that of [26], we recall its main lines and explain several technical differences. The analysis of the equation begins by a reduction to the order one :
(i∂t+
√
∆2+ m2)u = wn+1, w := (∆2+ m2)−1
2Re(u), (4)
where we have introduced u := (−i∂t+
√
∆2+ m2)w (remember that w takes real values). The problem
is equivalent to get a priori bounds on ||u(t)||Hs once we assumed the initial condition ||u(0)||Hs is of
order ε. One easily proves the estimate dtd||u(t)||2
Hs = O ||wn+1||Hs||u(t)||Hs that leads to the a priori
bound dtd||u(t)||2
Hs= O ε2+n. An integration around t = 0 merely leads to the local existence time ε−n.
Let us roughly explain the strategy of the normal form to improve the local existence time (we also refer the reader to [20, pages 541-544]).
Using (4) and decomposing wn+1 with the spectral projectors of −∆, we will construct four (n + 2)-multilinear operator M`, cM`, R, bR` such that the derivative of the squared Sobolev norm ||u(t)||2Hs can
be written as a sum n X `=0 Re ihM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ihR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui (5) + n X `=0 Re ih cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ih bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui.
A precise statement is given in Proposition 4.1. The idea is then to eliminate cM`, R, bR`by adding a higher
order term M (u(t)) of ||u(t)||2
Hs. In other words, one has M (u) = o(ε2) and dtd ||u(t)||2Hs− M (u(t))
does not contain cM`, R, bR`. A similar strategy is used to eliminate a part of the term M` by a normal
form procedure (the author has not succeeded to totally eliminate this term and this explains why the improved time is less than ε−2n, see [20, page 542]). Such a strategy leads to get an a priori estimate of the form dtd ||u(t)||2Hs− M (u(t)) = O ε2+An with A > 1. Note now that ||u(t)||2Hsand ||u(t)||2Hs−M (u(t))
are both equivalent near 0 and are of order ε2. An integration around t = 0 leads to improve the local existence time to ε−An(see the licit computations in Section 5).
Let us now explain several differences with [26].
• In the normal form procedure, one meets the following “small divisors issue” : one needs to estimate Z-linear combinations of n + 2 eigenvalues p(λ2k− 1)2+ m2 of
√
∆2+ m2. Using the fact that r−2 is
Diophantine and the asymptotic λk '
√
k, we get for every m > 0 and ℵ > 1 ∀k1> k2 q (λ2 k1− 1) 2+ m2−q(λ2 k2− 1) 2+ m2≥ C(m, r, ℵ) (λk1+ λk2) 2ℵ.
Following a proof by Zhang [26], we will prove that for any positive number ρ > 0, for almost all positive number m > 0 (in the sense of Lebesgue), there is a real number ν0= ν0(n, r, ρ, m) > 0 such that for any
(k0, . . . , kn+1) ∈ (N\{0})n+2the following bound from below holds true :
n+1 X `=0 ±q(λ2 k` − 1) 2+ m2 ≥ C(m, n, r, ρ) (λk0+ λkn+1) 4+ρmax(λ k1, . . . , λkn) ν0. (6)
except of course if the left-hand side of (6) vanishes. Such an inequality is useful to eliminate a part of the nonlinearity wn+1in the beam equation (1). In the papers [26, 11], one has the inclusion
{λk, k ≥ 1} ⊂ {
√
k, k ≥ 1}, (7)
which implies that λ2k+1− λ2
k is bounded from below. In our work, we merely has the weaker estimate
(see Lemma 2.1)
∀ℵ > 1 ∀k ≥ 1 λ2k+1− λ2k≥ C(ℵ)k−ℵ (8)
in mind the following rule : the smaller the exponent of (λk0+ λkn+1) is, the better the improvement of
the local existence time can be.
• The second point we want to stress is the reason why we are not able to deal with the Klein-Gordon equation on an irrational torus S1× rS1. To understand the interaction of the nonlinearity wn+1with the
linear part, the normal form procedure needs to prove that some multilinear operators, used to eliminate M`, cM`, R, bR` (see (5)), are bounded on the Sobolev space Hs(S1× rS1). In [26, line (2.1.9)], the
following estimates, proven thanks to (7), are used to get the boundedness on the Sobolev spaces :
∀ω > 2 ∀a ≥ 1 ∀` ≥ 1 X k≥1 1 (|λk− λ`| + a)ω ≤ Cλ` aω−2. (9)
To our knowledge, it is not clear whether a sequence (λk)k≥1which is badly separated in the sense of (8)
can satisfy (9). In our work, Lemma 3.4 will only give the weaker version
∀ω > 2 ∀a ≥ 1 ∀` ≥ 1 X k≥1 1 (|λk− λ`| + a)ω ≤ Cλ 2 ` aω−2. (10)
This simple fact unfortunately forbids in our approach to consider the Klein-Gordon equation. However, the beam equation admits a regularizing effect that allows us to deal with this issue. Let us recall that in the framework of the Klein-Gordon equation (∂2
t − ∆ + m2)w = wn+1 with u replaced by
(−i∂t+
√
−∆ + m2)w, the nonlinearity becomes h(−∆ + m2)−1 2Re(u)
in+1
. For the beam equation, the fact that the operator (∆2+ m2)−1
2 is more regularizing than (−∆ + m2)−12 gives a little gain of
derivatives which counterbalances the multiplicative lost λ2
` in (10). Consequently we will be able to
eliminate, totally or partially, the operators M`, cM`, R, bR`. We finally conclude by obtaining better a
priori estimates.
In Section 2, we sum up the spectral analysis we need for our purpose (the asymptotic behavior of the eigenvalues, the multilinear estimates of the spectral projectors and the small divisors estimates proved in Section 6). Sections 3 and 4 are devoted to the analysis of the nonlinearity with the aid of specific multilinear operators (we follow the same scheme of proof than that of [26] and thus we skip several similarities). Theorem 1.2 is proven in Section 5.
Section 3 Proposition 4.1 @ @ @ @ @ @ @ R -- -? Corollary 2.3
Lemma 2.1 (proved in Section 6)Proposition 2.4 Section 5
To make shorter several formulas, it will be convenient to write Hsinstead of Hs(S1× rS1).
2
Spectral analysis
2.1
Spectrum and universal multilinear estimates
In all the paper, we assume that r−2 belongs to the set D (see Definition 1.1). The spectrum of the operator −∆ on L2
(S1× rS1) is pure point and is given by
Sp(−∆) = {p2+ r−2q2, (p, q) ∈ Z2} = {p2+ r−2q2
We sort without counting multiplicities Sp(−∆ + 1) as an increasing sequence (λ2k)k≥1with λk≥ 0. For
instance, one has λ0= 1.
Lemma 2.1. Fix any σ > 32 and ℵ > 1, the following asymptotics hold as k tends to infinity : λk ' √ k, (11) λk+1− λk & k−σ, (12) λ2k+1− λ2 k & k −ℵ, (13)
where the symbols & and ' involve constants which may depend on (σ, ℵ, r).
Proof. The proof of (11) uses a basic Weyl law argument. We begin by writing for any positive and real number N : #{(p, q) ∈ N2, p2+ r−2q2≤ N } − br√N c X q=0 1 + bpN − r−2q2c = 0 #{(p, q) ∈ N2, p2+ r−2q2≤ N } − N r × 1 r√N br√N c X q=0 s 1 − q r√N 2 ≤ C(r)√N .
Recognizing a modified Riemann sum and using the irrationality of r−2, we get the following asymptotics as N tends to infinity : #{(p, q) ∈ N2, p2+ r−2q2≤ N } ∼ N r Z 1 0 p 1 − γ2dγ ∼ rπ 4 N #{k ∈ N?, λ2k− 1 ≤ N } ∼ rπ 4 N (14) k ∼ rπ 4 λ 2 k.
To see (12) and (13), we write
λ2k+1= 1 + P2+ r−2Q2> λ2k= 1 + p2+ r−2q2 where P, Q, p and q are integers. If q = Q holds, then λ2k+1− λ2
k ≥ 1 obviously holds and we get
λk+1− λk &√1k thanks to (11). If q 6= Q holds, we remember Definition 1.1 and we get for any µ > 2
λ2 k+1− λ2k = (P2− p2) + r−2(Q2− q2) & |Q2− q12|µ−1 ≥ 1 |Q2+ q2|µ−1 & 1 λ2µ−2k+1 , λk+1− λk & 1 λ2µ−1k & 1 kµ−1 2 .
2.2
Multilinear estimates
For any integer k ≥ 1, let us denote by Πk the spectral projector of L2(S1× rS1) on ker(1 − ∆ − λ2k).
We also denote by D(S1
× rS1
) the vector space of smooth functions on S1
× rS1. We will make use
Proposition 2.2. Consider two integers n ≥ 2 and N ≥ 1. For any real number ν > n, there is a positive constant C(N, ν, n, r) such that for any u0, . . . , un+1 ∈ D(S1× rS1), any nonnegative integers
k1≤ · · · ≤ kn+1≤ k0 and N ∈ N? one has
Z S1×rS1 Πk0(u0) . . . Πkn+1(un+1)dxdy ≤ C(N, ν, n, r)λν kn 1 +λk0− λkn+1 λkn −N n+1 Y j=0 ||uj||L2(S1×rS1), (15) where dxdy is the Riemannian measure on S1× rS1.
Proof. It is clear that it suffices to assume that each uj belongs to the range of Πkj for any integer
j ∈ [0, n + 1]. By writing λ2
kj− 1 = p
2
j+ r−2qj2with (pj, qj) ∈ N2and remembering that r−2 is irrational,
we can write uj(x, y), with (x, y) ∈ R2/(2πZ × 2πrZ), as a sum of at most four simple functions of the
form exp ±ipjx ± iqjyr and the left-hand side of (15) is a sum of at most 4
n+2integrals of products of
the above trigonometric functions. Thus, if the left-hand side of (15) is not zero then there are numbers τ0, τ00, . . . , τn+1, τn+10 ∈ {±1} such that
Z
S1×rS1
expi(τ0p0+ · · · + τn+1pn+1)x + i(τ00q0+ · · · + τn+10 qn+1)
y r
dxdy 6= 0
In other words, we have
τ0p0+ · · · + τn+1pn+1 = 0 τ00q0+ · · · + τn+10 qn+1 = 0. In particular, we deduce |p0− pn+1| + |q0− qn+1| ≤ n X j=1 pj+ qj≤ C(r)nλkn.
Hence, using the property that (x, y) 7→p1 + x2+ r−2y2 is a Lipschitz function on [0, +∞)2, we get
λk0− λkn+1 ≤ C(r)nλkn. (16)
It is now easy to prove (15) by using a H¨older inequality and the Sobolev embedding Hnν(S1× rS1) ⊂
L∞(S1× rS1). Indeed, the left-hand side of (15) is less than or equal to
||u0||L2(S1×rS1)||un+1||L2(S1×rS1) n Y j=1 ||uj||L∞(S1×rS1)≤ C ν n, r n λνkn n+1 Y j=0 ||uj||L2(S1×rS1).
The conclusion comes from (16).
In the same spirit that in the paper [26], we will indeed use the following corollary. Corollary 2.3. Let us consider δ ∈ (0, 1) such that 1δ − δ2< 1 and 1−δ2
δ < 1 hold
1. Consider moreover
two integers n ≥ 1 and N ≥ 1. For any real number ν > n, there is a positive number C = C(N, ν, n, r, δ) such that for any (k0, . . . , kn+1) ∈ (N\{0})n+2with max(k1, . . . , kn) ≤ kn+1and for any (u1, . . . , un+1) ∈
D(S1
× rS1)n+1 the following two assertions hold
i) if λk0
λkn+1 ∈δ, 1
δ holds then one has
||Πk0 Πk1(u1) . . . Πkn+1(un+1) ||L2(S1×rS1) ≤ C max(λk1, . . . , λkn) ν 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=1 ||uj||L2(S1×rS1). (17)
ii) if λk0
λkn+1 6∈δ, 1
δ holds or if max(λk1, . . . , λkn) > δ
2λ
kn+1 holds, then one has
||Πk0 Πk1(u1) . . . Πkn+1(un+1) ||L2(S1×rS1)≤ C max2(λk1, . . . , λkn+1) ν+N (λk0+ · · · + λkn+1) N n+1 Y j=1 ||uj||L2(S1×rS1), (18)
where max2(λk1, . . . , λkn+1) is the second largest number among λk1, . . . , λkn+1.
Proof. i) We begin by assuming k0 ≤ max(k1, . . . , kn) ≤ kn+1. The third largest number among
λk0, . . . , λkn+1 is of the same order than λkn+1, λk0 and λmax(k1,...,kn). Thus, we can write
|λk0− λkn+1| ≤ λk0+ λkn+1 ≤ 1 + 1 δ λk0 ≤ 1 + 1 δ max(λk1, . . . , λkn). (19) We can bound1 + λk0−λλ kn+1 kn −N by 1 in (15) and we get ||Πk0 Πk1(u1) . . . Πkn+1(un+1) ||L2(S1×rS1) = sup u06=0 1 ||u0||L2(S1×rS1) Z X u0Πk0 Πk1(u1) . . . Πkn+1(un+1) dx = sup u06=0 1 ||u0||L2(S1×rS1) Z X Πk0(u0) . . . Πkn+1(un+1)dx (20) . max(λk1, . . . , λkn) ν n+1 Y j=1 ||uj||L2(S1×rS1).
Using (19), we obtain (17). If k0> max(k1, . . . , kn) holds, then (17) is nothing else than (15).
ii) Let us explain why there exists % ∈ (0, 1) (independent of k0, . . . , kn+1) such that the following
inequality is true :
max2(λk0, . . . , λkn+1) − max(λk1, . . . , λkn) ≤ % max(λk0, . . . , λkn). (21)
Note that the inequality 0 ≤ max2(λk0, . . . , λkn+1) − max(λk1, . . . , λkn) obviously holds. We now consider
several subcases : • if λk0
λkn+1 > 1
δ holds then we have max2(λk0, . . . , λkn+1) = λkn+1 and max(λk0, . . . , λkn+1) = λk0. The
number % = δ is convenient. • if λkn+1λ
k0 >
1
δ holds then we have max(λk0, . . . , λkn+1) = λkn+1. We should make a discussion about the
position of λk0. If λk0 ≤ max(λk1, . . . , λkn) ≤ λkn+1 holds then the left-hand side of (21) vanishes. If
max(λk1, . . . , λkn) ≤ λk0≤ λkn+1 then one just have to write
max2(λk0, . . . , λkn+1) − max(λk1, . . . , λkn) = λk0− max(λk1, . . . , λkn) ≤ λk0 ≤ δλkn+1.
• let us assume that λk0
λkn+1 belongs to [δ, 1
δ], max(λk1, . . . , λkn) > δ
2λ
kn+1 and max(λk1, . . . , λkn) ≤
λkn+1 ≤ λk0 hold. We then have
max2(λk0, . . . , λkn+1) − max(λk1, . . . , λkn) ≤ 1 − δ 2 λ kn+1 ≤ 1 δ 1 − δ 2 λ k0.
The previous gives (21) because we have assumed that 1−δδ2 is less than 1. • let us assume that λk0
λkn+1 belongs to [δ, 1
δ], max(λk1, . . . , λkn) > δ
2λ
kn+1 and max(λk1, . . . , λkn) ≤ λk0≤
λkn+1 hold. Thus, we get
• we finally assume that λk0 λkn+1 belongs to [δ, 1 δ], max(λk1, . . . , λkn) > δ 2λ kn+1and λk0 ≤ max(λk1, . . . , λkn) ≤
λkn+1 hold. Such a case is obvious because the left-hand side of (21) vanishes.
The inequality (21) is proven. Let us prove (18). We note now that we have max3(λk0, . . . , λkn+1) ≤
max2(λk1, . . . , λkn+1) = max(λk1, . . . , λkn) where max3(λk0, . . . , λkn+1) is the third largest number among
λk0, . . . , λkn+1. By combining with (21), that gives us
λk0+ · · · + λkn+1
max2(λk1, . . . , λkn+1)
. max(λk0, . . . , λkn+1)
max2(λk1, . . . , λkn+1)
. max(λk0, . . . , λkn+1) − max2(λk0, . . . , λkn+1) + max(λk1, . . . , λkn)
max2(λk1, . . . , λkn+1)
. 1 +max(λk0, . . . , λkn+1) − max2(λk0, . . . , λkn+1)
max3(λk0, . . . , λkn+1)
.
From (15) and (20), we get the conclusion.
2.3
Small divisors
For any m > 0 and ` ∈ [0, n] ∩ N, let us define the following two maps Fm` and bFm` on [1, +∞)2n+2 :
Fm`(ξ0, . . . , ξn+1) = ` X j=0 q (ξ2 j − 1)2+ m2− n+1 X j=`+1 q (ξ2 j− 1)2+ m2, b Fm`(ξ0, . . . , ξn+1) = ` X j=0 q (ξ2 j − 1)2+ m2− n X j=`+1 q (ξ2 j− 1)2+ m2+ q (ξ2 n+1− 1)2+ m2. (22)
We also define some specific subsets of (N\{0})n+2 :
Ωn+2(`) := {(k0, . . . , kn+1), {k0, . . . , k`} = {k`+1, . . . , kn+1}},
Ωn+2(b`) := {(k0, . . . , kn+1), {kn+1, k0, . . . , k`} = {k`+1, . . . , kn}} .
(23)
Note that the previous definitions are relevant only if n is even. Note also that for any k ∈ Ωn+2(`)
one has Fm`(λk0, . . . , λkn+1) = 0. The purpose of the next result is to explain that, for a generic m > 0
and for any k 6∈ Ωn+2(`), the number |Fm`(λk0, . . . , λkn+1)| is bounded from below.
Proposition 2.4. For any positive number ρ > 0, for almost every m > 0 (in the sense of Lebesgue), any integer ` ∈ [0, n], there are C > 0 and ν0> 0 such that for all (k0, . . . , kn+1) ∈ (N\{0})n+2\Ωn+2(`)
we have the following estimates
1 |F` m(λk0, . . . , λkn+1)| ≤ C(λk0+ λkn+1) 4+ρ max(λk1, . . . , λkn) ν0, (24) 1 |F` m(λk0, . . . , λkn+1)| ≤ C(λk0+ · · · + λkn+1) ν0. (25)
In the same spirit, if (k0, . . . , kn+1) ∈ (N\{0})n+2\Ωn+2(b`) then we have
Remark 2.5. It is obvious that (26) is stronger than (27). However, the proof of (26) will be a conse-quence of (27).
Remark 2.6. We will see at the end of the proof that the number ρ > 0 in Proposition 2.4 is linked to the number A ∈ (1,54) of Theorem 1.2 by the formula A = 1 + 4+ρ1 .
The proof of Proposition 2.4 uses in an essential way Lemma 2.1 and is adapted from [26, Theorem 2.3.1 and Proposition 2.3.6] and [7, Part 2.1]. We postpone the proof in Section 6 because it is quite technical.
3
Multilinear operators
In this part, we merely use the asymptotic λk '
√
k. As usual, we respectively denote by D0(S1× rS1)
and L(D(S1× rS1), D0
(S1× rS1
)) the vector spaces of the distributions on S1× rS1 and of the linear
operators from D(S1× rS1) to D0
(S1× rS1). In the sequel we give sufficient conditions on n-multilinear
operators M : D(S1
× rS1)n
→ L(D(S1
× rS1), D0(S1
× rS1)) so that they admit a bounded extension
from (Hs (S1 × rS1))n to L(Hs (S1 × rS1), H−s0 (S1
× rS1)) for some real number s0.
Definition 3.1. Let us consider τ > 0, ν > 0, δ ∈ (0, 1) and n ∈ N?. We say that a multilinear operator M : D(S1
× rS1)n
→ L(D(S1
× rS1), D0
(S1× rS1)) belongs to Mτ,ν
n,δ if for every N > 1 one can find a
constant C > 0 such that for any (u1, . . . , un+1) ∈ D(S1× rS1)n+1 one has
i) if δ ≤ λk0
λkn+1 ≤ 1
δ and max(k1, . . . , kn) ≤ kn+1hold then
||Πk0 M(Πk1u1, . . . , Πknun)Πkn+1un+1 ||L2(S1×rS1) ≤ Cλτ k0max(λk1, . . . , λkn) ν 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=1 ||uj||L2(S1×rS1).
ii) for all other frequencies, one has Πk0 M(Πk1u1, . . . , Πknun)Πkn+1un+1 = 0.
The following proposition is proven below.
Proposition 3.2. Consider four positive numbers ν, δ, τ and s. There is s0 = s0(ν) > 0 such that any
M ∈ Mτ,νn,δ admits a unique extension as a bounded operator from (Hs)n to L(Hs, Hs−τ −2) for s > s 0.
That means that the following inequality holds for any u1, . . . , un+1∈ D(S1× rS1)
||M(u1, . . . , un)un+1||Hs−τ −2 ≤ C
n+1
Y
j=1
||uj||Hs.
Remark 3.3. In the paper [26], the sequence λk behaves as
√
k and λk+1− λk as
√
k + 1 −√k. This separation property allows to get the stronger estimate :
||M(u1, . . . , un)un+1||Hs−τ −1 ≤ C
n+1
Y
j=1
||uj||Hs.
From now, we will call the Hs-boundedness the property which holds for any M ∈ M2s−2,νn,δ in Proposition 3.2 : ∀(u0, u1, . . . , un+1) ∈ D(S1× rS1)n+2 |hM(u1, . . . , un)un+1, u0i| ≤ C n+1 Y j=0 ||uj||Hs. (28)
The power of λk0 in the bound of the following lemma is the essential reason which allows to prove
Lemma 3.4. For any integer k0≥ 1, any real numbers a ≥ 1 and ω > 2, one has X kn+1≥1 1 (|λkn+1 − λk0| + a) ω ≤ Cλ2k 0 aω−2,
where C > 0 is independent with respect to (a, k0).
Proof. We begin by separating N? in two subsets S(λk0) t T (λk0) :
S(λk0) := {kn+1≥ 1, λkn+1 ≤ 2λk0}
T (λk0) := {kn+1≥ 1, λkn+1 > 2λk0}.
From (11) (or (14)), we obviously have 1 + max S(λk0) = min T (λk0) ' λ
2 k0. So we can write X S(λk0) 1 (|λkn+1− λk0| + a) ω . λ2 k0 aω.
Note that the inequality λkn+1− λk0 >
1 2λkn+1 holds in T (λk0), we get X T (λk0) 1 (|λkn+1− λk0| + a) ω . X T (λk0) 1 (pkn+1+ a)ω . X kn+1≥2 1 (kn+1+ a2)ω/2 . Z +∞ 1 dx (x + a2)ω/2.
As ω is a fixed number greater than 2, the previous bound is less than or equal to a−2(ω2−1)= a−(ω−2),
up to a multiplicative constant. We easily conclude. Following the same lines than [26], we prove Proposition 3.2.
Proof. Let us define
Ω(δ) = n(k1, . . . , kn+1) ∈ (N?)n+1, δ ≤ λkn+1 λk0 ≤ 1 δ, max(k1, . . . , kn) ≤ kn+1 o , Ψ(δ) = {(k1, . . . , kn+1) ∈ Ω(δ), k1≤ · · · ≤ kn+1} ⊂ Ω(δ). (29)
The square of the norm ||M(u1, . . . , un)un+1||Hs−τ −2 is
X k0≥1 λ2(s−τ −2)k 0 ||Πk0(M(u1, . . . , un)un+1)|| 2 L2(S1×rS1) = X k0≥1 λ2(s−τ −2)k 0 X k1,...,kn+1 Πk0 M (Πk1u1, . . . , Πknun) Πkn+1un+1 2 L2(S1×rS1) ,
which is less than or equal to
C X k0≥1 λ2(s−τ −2)k 0 X Ω(δ) λτk0max(λk1, . . . , λkn) ν 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=1 ||Πkjuj||L2(S1×rS1) 2 .
The symmetry of the variables k1, . . . , kn+1allows to replace Ω(δ) by Ψ(δ). Thus, it suffices to bound
From the Cauchy-Schwarz inequality, we bound the previous term by C X k0≥1 λ2s−4k 0 Θ1Θ2 where Θ1 and Θ2are defined by Θ1:= X Ψ(δ) λνk n 1 +|λk0− λkn+1| λkn −N n Y j=1 ||Πkjuj||L2(S1×rS1), Θ2:= X Ψ(δ) λνk n 1 +|λk0− λkn+1| λkn −N ||Πkn+1un+1|| 2 L2(S1×rS1) n Y j=1 ||Πkjuj||L2(S1×rS1).
For any N = ω > 2, Lemma 3.4 allows us to bound
Θ1 = X Ψ(δ) λν+Nk n (λkn+ |λk0− λkn+1|) N n Y j=1 ||Πkjuj||L2(S1×rS1) ≤ X k1,...,kn λν+2k n λ 2 k0 n Y j=1 ||Πkjuj||L2(S1×rS1) ≤ λ2 k0 n Y j=1 X kj≥1 λν+2k j ||Πkjuj||L2(S1×rS1) ≤ λ2 k0 n Y j=1 v u u u t X kj≥1 λ−2(s−ν−2)k j X kj≥1 λ2s kj||Πkju|| 2 L2(S1×rS1) . λ2 k0 n Y j=1 ||uj||Hs,
provided that s 1 holds (thanks to (11)). We can go on and bound ||M(u1, . . . , un)un+1||2Hs−τ −2 by
X k0≥1 λ2s−4k 0 × Θ2× λ 2 k0 n Y j=1
||uj||Hs, which is nothing else than
n Y j=1 ||uj||Hs X k0≥1 Ψ(δ) λ2s−2k 0 λ ν kn 1 + |λk0− λkn+1| λkn −N ||Πkn+1un+1|| 2 L2(S1×rS1) n Y j=1 ||Πkjuj||L2(S1×rS1).
We now have to use the estimate λkn+1 ' λk0 in the set Ψ(δ) (see (29)) to get the bound
n Y j=1 ||uj||Hs X k0≥1 Ψ(δ) λ2s−2k n+1λ ν+N kn (λkn+ |λk0− λkn+1|) N||Πkn+1un+1|| 2 L2(S1×rS1) n Y j=1 ||Πkjuj||L2(S1×rS1).
Still using Lemma 3.4 (and inverting k0and kn+1), we can bound
n Y j=1 ||uj||Hs X Ψ(δ) λ2skn+1λν+2k n ||Πkn+1un+1|| 2 L2(S1×rS1) n Y j=1 ||Πkjuj||L2(S1×rS1) ≤ C n Y j=1 ||uj||Hs ||un+1||2Hs X k1,...,kn n Y j=1 λν+2k j ||Πkjuj||L2(S1×rS1),
which gives the conclusion as above.
Definition 3.5. Let M be an operator in Mτ,νn,δ and ` be an integer in [0, n]. We write M ∈ Mτ,νn,δ[`] if for all (k0, . . . , kn+1) ∈ (N\{0})n+2 and u1, . . . , un+1∈ D(S1× rS1) we have
{k0, . . . , k`} = {k`+1, . . . , kn+1} ⇒ Πk0 M(Πk1u1, . . . , Πknun)Πkn+1un+1 = 0.
We also write M ∈ Mτ,νn,δ[b`] if the previous implication is replaced by the following one {kn+1, k0, . . . , k`} = {k`+1, . . . , kn} ⇒ Πk0 M(Πk1u1, . . . , Πknun)Πkn+1un+1 = 0.
Let us define other multilinear operators.
Definition 3.6. Consider τ ∈ R, ν > 0 and an integer ` ∈ [0, n]. A multilinear operator R : D(S1× rS1)n
→ L(D(S1
× rS1), D0(S1
× rS1)) is in the class Rτ,ν
n [`] if the following two properties hold
i) for any N ≥ 1 there is C > 0 such that for any (k0, . . . , kn+1) ∈ (N\{0})n+2, u1, . . . , un+1 ∈
D(S1
× rS1) the L2-norm ||Π
k0(R(Πk1u1, . . . , Πknun)Πkn+1un+1)||L2(S1×rS1) is less than or equal to
Cλτk0max2(λk1, . . . , λkn+1) ν+N (λk0+ · · · + λkn+1) N n+1 Y j=1 ||uj||L2(S1×rS1),
ii) for all (k0, . . . , kn+1) ∈ (N\{0})n+2 and u0, . . . , un+1∈ D(S1× rS1) we have
{k0, . . . , k`} = {k`+1, . . . , kn+1} ⇒ Πk0 R(Πk1u1, . . . , Πknun)Πkn+1un+1 = 0.
We also write R ∈ Rτ,ν
n [b`] if i) holds and if the second condition ii) is replaced by
{kn+1, k0, . . . , k`} = {k`+1, . . . , kn} ⇒ Πk0 R(Πk1u1, . . . , Πknun)Πkn+1un+1 = 0.
As for the space Mτ,νn,δ, we have a Hs-boundedness property for the spaces Rτ,νn [`] and Rτ,νn [b`].
Proposition 3.7. Consider ν, τ, s > 0, there is s0 = s0(ν) > 0 such that if s and 3s − τ are larger
than s0(ν) then any operator R ∈ Rτ,νn [`] ∪ Rτ,νn [b`] admits a unique extension as a bounded operator from
(Hs
(S1× rS1))n to L(Hs
(S1× rS1), H−s
(S1× rS1)). In other words, the following inequality holds for
all u1, . . . , un+1∈ D(S1× rS1) ||R(u1, . . . , un)un+1||H−s(S1×rS1)≤ C n+1 Y j=1 ||uj||Hs(S1×rS1).
Proof. The proof is the same than in [26, Proposition 2.1.5] and uses the mere asymptotic property λk'
√
k (see (11)).
4
Towards the regularizing effect of the beam equation
The scope of this part is to explain the proof of Proposition 4.1 that will be used in the next parts. Remember the reduction u := (−i∂t+
√
∆2+ m2)w we made to get (4). Defining Λ m:=
√
∆2+ m2, we
have
||Λs/2m u||L2(S1×rS1)' ||u||Hs(S1×rS1)' ||∂tw||Hs(S1×rS1)+ ||w||Hs+2(S1×rS1).
Thus, if we introduce the squared Sobolev norm Θs(u) := 1 2||Λ s 2 mu||2L2(S1×rS1),
then Theorem 1.2 is equivalent to the proof of the following a priori estimates for all small enough number ε > 0 :
∀t ∈ (−Cε−An, +Cε−An) Θs(u(t)) ≤ Kε2,
Proposition 4.1. There is a number δ ∈ (0, 1) such that the following holds. For each integer ` ∈ [0, n], there are M`∈ M2s−3,νn,δ [`], Md`∈ M2s−2,νn,δ [b`], R`∈ R2s,νn [`], Rb`∈ R2s,νn [b`] such that d dtΘs(u(t)) = n X `=0 Re ihM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ihR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ih cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ih bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui.
The proof is quite similar to that of [26, Proposition 2.2.1] (see also a slightly modified version in [20, Proposition 4.1]) but the exponents are different. Therefore, we will write a proof which only focus on the reason why the regularizing effect of the beam equation allows for the exponent 2s − 3 for M` and skip
the construction of cM`, R` and bR`. We only recall that the construction of the previous operators rely
on the multilinear estimates (15). More precisely, M` and cM` are constructed thanks to (17) whereas
R` and bR` are constructed thanks to (18).
We stress that it is very important that the exponent 2s − 3 is less than 2s − 2 given by the Hs
-boundedness (see (28)). As shortly explained in the introduction, this is the issue we have not overcome to deal with the Klein-Gordon equation on a Diophantine irrational torus S1× rS1.
Repeating the same reasoning as in [26], there is a bounded function b : (N\{0})n+1→ R which has support in Γ := {(k1, . . . , kn+1) ∈ (N?)n+1, max(k1, . . . , kn) ≤ kn+1}, and satisfies wn+1 = X k1,...,kn+1≥1 Πk1(w) . . . Πkn+1(w) = X Γ b(k1, . . . , kn+1) n+1 Y j=1 Πkj(w). (30)
Furthermore, the map b is constant on the subset of elements (k1, . . . , kn+1) ∈ Γ such that max(k1, . . . , kn) <
kn+1. Remember that w and u are related by the equality
w = Λ−1m u + u 2 =1 2Λ −1 mu + 1 2Λ −1 mu.
Putting the above expression in (30) leads us to the following formula wn+1= − n X `=0 C`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u − n X `=0 C`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u,
where C` is given for any f ∈ D(S1× rS1) by
for some constant Kn,` which is a function of n and `. Using ˙u = iΛmu − iwn+1 (see (4)), we easily get d dtΘs(u(t)) = RehΛ s 2 mu, Λ˙ s 2 mui = −Re ihΛ s 2 mwn+1, Λ s 2 mui + =0 z }| { Re ihΛs+12 u, Λ s+1 2 ui = −Re ihΛs2 mwn+1, Λ s 2 mui = n X `=0 Re ihΛs2 mC`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, Λs2 mui + n X `=0 Re ihΛ s 2 mC`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, Λ s 2 mui.
The following two lemmas will entirely prove Proposition 4.1.
Lemma 4.2. There is a number δ ∈ (0, 1) such that the following holds. For each integer ` ∈ [0, n], there are M`∈ M2s−3,νn,δ [`] and R`∈ R2s,νn [`] such that
n X `=0 Re ihC`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, Λsmui = n X `=0 Re ihM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ihR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui.
Lemma 4.3. There is a number δ ∈ (0, 1) such that the following holds. For each integer ` ∈ [0, n], there are dM`∈ M2s−2,νn,δ [b`] and bR`∈ R2s,νn [b`] such that
n X `=0 Re ihC`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, Λsmui = n X `=0 Re ih cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + n X `=0 Re ih bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui.
Lemma 4.3 is similar to [26, Lemma 2.2.3]. As written above, we merely prove the part of Lemma 4.2 which involves M`. As in [26, page 648], we only need to consider a part of C`, called below C`,1,
which contains frequencies such that λk0 ' λkn+1 and max(λk1, . . . , λkn) . λkn+1. More precisely, we
will consider C`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )f := Kn,` X Υ b(k1, . . . , kn+1) Qn+1 j=1 q (λ2 kj − 1) 2+ m2 Πk0 ` Y j=1 Πkj(u) n Y j=`+1 Πkj(u) Πkn+1(f ) ,
where Υ is defined, for some δ ∈ (0, 1), by
Υ := (k0, . . . , kn+1) ∈ (N?)n+2, δ ≤ λkn+1 λk0 ≤1 δ, max(λk1, . . . , λkn) ≤ δ 2λ kn+1 . (31)
Let us add that the contribution of C` which is parametrized by the complementary subset of Υ will
indeed contribute in R`. Let us begin with the following equality :
Re ihC`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−`
)u, Λsmui = −Re ihC`,1(u, . . . , u
Introducing the commutator [Λsm, C`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )], we get 2Re ihΛs2 mC`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, Λs2 mui = Re ihΛ s mC`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−`
)u, ui − ihC`,1(u, . . . , u
| {z } ` , u, . . . , u | {z } n−` )?Λsmu, ui = Re ihC`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )Λsmu − C`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )?Λsmu, ui (32)
+Re ih[Λsm, C`,1(u, . . . , u
| {z } ` , u, . . . , u | {z } n−` )]u, ui. (33)
Let us handle (32). Permuting k0and kn+1, we get that C`,1(u, . . . , u
| {z }
`
, u, . . . , u | {z }
n−`
)?f is nothing else than
Kn,` X Υ? b(k1, . . . , kn, k0) Qn j=0 q (λ2 kj − 1) 2+ m2) Πk0 Πkn+1(f ) ` Y j=1 Πkj(uj) n Y j=`+1 Πkju , Υ?:= (k0, . . . , kn+1) ∈ (N?)n+2, δ ≤ λkn+1 λk0 ≤ 1 δ, max(λk1, . . . , λkn) ≤ δ 2λ k0 . Let us define B(k0, . . . , kn+1) := b(k1, . . . , kn+1) q (λ2 kn+1− 1) 2+ m2) −qb(k1, . . . , kn, k0) (λ2 k0− 1) 2+ m2) .
This definition leads us to reformulate (32) as
Kn,`Re i X Υ∩Υ? B(k0, . . . , kn+1) ((λ2 kn+1− 1) 2+ m2)s 2 Qn j=1 q (λ2 kj− 1) 2+ m2) * ` Y j=1 Πkju n+1 Y j=`+1 Πkju, Πk0u + +Kn,`Re i X Υ\(Υ∩Υ?) b(k1, . . . , kn+1) ((λ2 kn+1− 1) 2+ m2)s 2 Qn+1 j=1 q (λ2k j − 1) 2+ m2 * ` Y j=1 Πkj(u) n+1 Y j=`+1 Πkj(u), Πk0(u) + −Kn,`Re i X Υ?\(Υ∩Υ?) b(k1, . . . , kn, k0) ((λ2 kn+1− 1) 2+ m2)s 2 Qn j=0 q (λ2 kj − 1) 2+ m2 * ` Y j=1 Πkj(u) n+1 Y j=`+1 Πkj(u), Πk0(u) + .
The definition (31) of Υ ensures that max(k1, . . . , kn) < min(k0, kn+1) holds. The properties of b we
recalled above (b is bounded and b(k1, . . . , kn, kn+1) takes a fixed constant if max(k1, . . . , kn) < kn+1
holds, see (30)) allows for the following bounds
∀k ∈ Υ ∩ Υ? |B(k 0, . . . , kn+1)| . 1 q (λ2 k0− 1) 2+ m2 −q 1 (λ2 kn+1− 1) 2+ m2 . |λk0− λkn+1| λ3 kn+1 (34) ∀k ∈ Υ\(Υ ∩ Υ?) q|b(k1, . . . , kn+1)| (λ2 kn+1− 1) 2+ m2 . 1 λ2 kn+1 .max(λk1, . . . , λkn) λ3 kn+1 (35) ∀k ∈ Υ?\(Υ ∩ Υ?) |b(k1, . . . , kn, k0)| q (λ2 k0− 1) 2+ m2 . 1 λ2 k0 . max(λk1, . . . , λkn) λ3 kn+1 . (36) Remembering (23), we define B(k0, . . . , kn+1) := B(k0, . . . , kn+1)1Υ∩Υ?+ b(k1, . . . , kn+1) q (λ2 kn+1− 1) 2+ m2 1Υ\(Υ∩Υ?)− b(k1, . . . , kn, k0) q (λ2 k0− 1) 2+ m2 1Υ\(Υ∩Υ?), M`,1(u1, . . . , un)un+1:= Kn,` X (N?)n+2\Ω n+2(`) B(k0, . . . , kn+1) ((λ2 kn+1− 1) 2+ m2)s 2 Qn j=1 q (λ2 kj− 1) 2+ m2) Πk0 n+1 Y j=1 Πkj(uj) . Removing Ωn+2(`) allows to prove, after an easy computation, that (32) is equal to
Re ihM`,1(u, . . . , u | {z } `times , u, . . . , u | {z } n−`times )u, ui.
The operator M`,1 is moreover nonresonant in the sense of Definition 3.5. It is also easy to prove that
M`,1 belongs to M2s−3,ν+2n,δ for some constant ν > n with the aid of the estimates (17) and by noticing
that a common upper bound of (34),(35) and (36) is max(λk1, . . . , λkn)λ
−3 kn+1(1 + |λk0− λkn+1|) : for any N ∈ N? we can write B(k0, . . . , kn+1) ((λ2 kn+1− 1) 2+ m2)s 2 Qn j=1 q (λ2 kj− 1) 2+ m2) Πk0 n+1 Y j=1 Πkj(uj) L2(S1×rS1) . λ2s−3k0 (1 + |λk0− λkn+1|) max(λk1, . . . , λkn) ν+1 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=1 ||uj||L2(S1×rS1) . λ2s−3k0 max(λk1, . . . , λkn) ν+2 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −(N −1) n+1 Y j=1 ||uj||L2(S1×rS1).
A similar strategy is possible to handle (33). From the definition of C`,1 (see (31) above), the
commutator takes the form h[Λs m, C`,1(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )]u, ui = Kn,` X Υ b(k1, . . . , kn+1)cm,s(k0, kn+1) n+1 Y j=1 q (λ2k j − 1) 2+ m2 * ` Y j=1 Πkj(u) n+1 Y j=`+1 Πkj(u), Πk0(u) + , with cm,s(k0, kn+1) := ((λ2k0− 1) 2+ m2)s 2− ((λ2 kn+1− 1) 2+ m2)s
2. Since b is bounded, we obviously have
We can finish as above.
The operators dM` and bR` come from Corollary 2.3 and considerations on the frequencies which
respectively satisfy λkn+1 ' λk0 and λkn+1 6' λk0. Following [26, page 652], similar ideas can be used to
handle cM` and we check that the following operator is convenient
c M`(u1, . . . , un)un+1:= Kn,` X max(k1,...,kn)≤kn+1 δλk0≤λkn+1≤1 δλk0 (k0,...,kn+1)6∈Ωn+2(b`) b(k1, . . . , kn+1)((λ2k0− 1) 2+ m2)s 2 Qn+1 j=1 q (λ2 kj+ m 2)2+ m2 Πk0 n+1 Y j=1 Πkj(uj) .
5
Proof of Theorem 1.2
The main result of the paper, namely Theorem 1.2, will be a consequence of Proposition 5.1 and 5.2. From now, we consider the operators M`∈ M2s−3,νn,δ [`], cM`∈ M2s−2,νn,δ [b`],R`∈ R2s,νn [`] and bR`∈ R2s,νn [b`]
in Proposition 4.1. We choose s large enough such that one can use Proposition 3.2 and Proposition 3.7. We also assume m > 0 to be generic in the sense of Proposition 2.4.
Proposition 5.1. There are a number ν0> 0 and three multilinear operators
c M`∈ M
2s−3,ν+ν0
n,δ [b`], R`∈ R2s,ν+νn 0[`], Rb`∈ R2s,ν+νn 0[b`]
such that for any solution u of ˙u = iΛmu − iwn+1, one has d
dth cM`(u, . . . , u, u, . . . , u)u, ui = h cM`(u, . . . , u, u, . . . , u)u, ui + O(||u|| 2n+2 Hs ),
d
dthR`(u, . . . , u, u, . . . , u)u, ui = hR`(u, . . . , u, u, . . . , u)u, ui + O(||u|| 2n+2 Hs ), d dth bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui = h bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + O(||u||2n+2Hs ).
Proof. We merely consider cM` because the other terms are similar. Since cM` is nonresonant (see
Definition 3.5), the following equality holds for any (u0, . . . , un+1) ∈ D(S1× rS1)n+2:
h cM`(u1, . . . , un)un+1, u0i = X max(k1,...,kn)≤kn+1 δλk0≤λkn+1≤1 δλk0 (k0,...,kn+1)6∈Ωn+2(b`) h cM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i.
Define now the operator cM`by
From (26), we get for any fixed integer N ∈ N? : h cM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i i bF` m(λk0, . . . , λkn+1) .λ 2s−2 k0 max(λk1, . . . , λkn) ν+ν0 λk0+ λkn+1 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=0 ||uj||L2(S1×rS1) . λ2s−3k0 max(λk1, . . . , λkn) ν+ν0 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=0 ||uj||L2(S1×rS1) ⇒ Πk0Mc`(Πk1u1, . . . , Πknun)Πkn+1un+1 L2(S1×rS1) . λ2s−3k0 max(λk1, . . . , λkn) ν+ν0 1 + |λk0− λkn+1| max(λk1, . . . , λkn) −N n+1 Y j=1 ||uj||L2(S1×rS1).
In other words, cM` belongs to M2s−3,ν+νn,δ 0[b`]. Using the relations ˙u = −iw
n+1+ iΛ
mu and ΛmΠk =
p(λ2
k− 1)2+ m2Πk, we can distribute iΛmu and leave the terms involving −iwn+1in a rest called below
Q(wn+1) : d dth cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−`
)u, ui = −ihM`(u, . . . , u
| {z } ` , u, . . . , u | {z } n−` )u, Λmui −i ` X k=1 h cM`(u, . . . , u | {z } k−1 , Λmu, u, . . . , u | {z } `−k , u, . . . , u | {z } n−` )u, ui +i n X k=`+1 h cM`(u, . . . , u | {z } ` , u, . . . , u | {z } k−`−1 , Λmu, u, . . . , u | {z } n−k )u, ui
−ih cM`(u, . . . , u, u, . . . , u)Λmu, ui + Q(wn+1)
= h cM`(u, . . . , u, u, . . . , u)u, ui + Q(wn+1).
The contribution Q(wn+1) contains n + 2 terms which involve wn+1, for instance
h cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, wn+1i.
Since 2s − 3 is less than 2s − 2, one can use Proposition 3.2 to see that cM` is bounded from (Hs)n to
L(Hs, H−s). The inequalities ||wn+1||
Hs . ||w||n+1
Hs ≤ ||u||
n+1
Hs leads us to the conclusion. The same
strategy is possible to deal with the other terms. Indeed, we have to make use of the other small divisors estimates (25), (27) and the Hs-boundedness of the families of operators R2s,ν
n [`] and R2s,νn [b`] family (see
Proposition 3.7).
Looking at Proposition 4.1, the previous result means that one can eliminate the three terms which involve cM` ∈ M2s−2,νn,δ [b`],R` ∈ Rn2s,ν[`] and bR` ∈ R2s,νn [b`]. If one uses a similar strategy for M` ∈
M2s−3,νn,δ [`], we encounter a little issue since the lost of 4 + ρ derivatives given by (24) is too bad, and thus we would obtain an operator which belongs to M2s−3+4+ρ,νn,δ [`] = M2s+1+ρ,νn,δ [`]. This is useless for us since we do not know if the previous space has the Hs-boundedness (see (28)). Therefore, a way to
overcome this issue is to eliminate only a part of M` (to our knowledge, such a strategy appears for the
first time in [7]). Let us consider a function ψ : (0, 1) → R+ such that lim
ε→0ψ(ε) = +∞ holds. We have
M`,ε and V`,ε : hM`,ε(u1, . . . , un)un+1, u0i := X Υ\Ωn+2(`) λk0<ψ(ε) hM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i, hV`,ε(u1, . . . , un)un+1, u0i := X Υ\Ωn+2(`) λk0≥ψ(ε) hM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i.
We will eliminate the M`,ε part and keep the one involving V`,ε. Remember that the Hs-boundedness
holds for M2s−2,νn,δ and that M`, M`,εand V`,εbelong to M 2s−3,ν
n,δ . Thus, V`,εfulfills the Hs-boundedness
uniformly in ε : |hV`,ε(u1, . . . , un)un+1, u0i| ≤ C n+1 Y j=0 ||uj||Hs.
Nevertheless, it is more interesting to take account the Hs-boundedness with the inequality λ2s−3k
0 ≤
λ2s−2
k0
ψ(ε) .
And we get the stronger bound
|hV`,ε(u1, . . . , un)un+1, u0i| ≤ C ψ(ε) n+1 Y j=0 ||uj||Hs. (37)
The term M`,εwill be eliminated by a normal form procedure.
Proposition 5.2. For any ρ > 0 and ε > 0, there are ν0 > 0 and M`,ε ∈ M2s−2,ν+νn,δ 0[`] such that for
any solution u of ˙u = iΛmu − iwn+1, one has
d
dthM`,ε(u, . . . , u, u, . . . , u)u, ui = hM`,ε(u, . . . , u, u, . . . , u)u, ui + ψ(ε)
3+ρO(||u||2n+2 Hs ).
Proof. For any (u0, . . . , un+1) ∈ D(S1× rS1)n+2, we define
hM`,ε(u1, . . . , un)un+1, u0i := X Υ\Ωn+2(`) λk0<ψ(ε) hM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i −iF` m(λk0, . . . , λkn+1) .
From (24), the division by the small divisor gives a lost of 4 + ρ powers of λk0+ λkn+1 which is similar
to λk0 (because we work in Υ). Consequently, we have
hM`(Πk1u1, . . . , Πknun)Πkn+1un+1, Πk0u0i −iF` m(λk0, . . . , λkn+1) ≤ Cλ(2s−3)+(4+ρ)k 0 max(λk1, . . . , λkn) ν max(λk1, . . . , λkn) N +ν0 (|λk0− λkn+1| + max(λk1, . . . , λkn)) N ≤ Cψ(ε)3+ρλ2s−2 k0 max(λk1, . . . , λkn) ν max(λk1, . . . , λkn) N +ν0 (|λk0− λkn+1| + max(λk1, . . . , λkn)) N.
Therefore, the Hs-boundedness property gives
|hM`,ε(u1, . . . , un)un+1, u0i| ≤ Cψ(ε)3+ρ n+1
Y
j=0
||uj||Hs. (38)
We easily conclude by computing d
The proof of Theorem 1.2 is a combination of the previous results. We write it for the convenience of the reader. Let us consider u a solution of ˙u = iΛmu − iwn+1for s large enough and define
Ms,ε(u) := Re i n X `=0 hM`,ε(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + h cM`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui +hR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui + h bR`(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui .
The inequality (37), Proposition 4.1, Proposition 5.1 and Proposition 5.2 give us d dt[Θs(u) − Ms,ε(u)] = ψ(ε) 3+ρO(||u||2n+2 Hs ) + n X `=0 Re iV`,ε(u, . . . , u | {z } ` , u, . . . , u | {z } n−` )u, ui = ψ(ε)3+ρO(||u||2n+2 Hs ) + 1 ψ(ε)O(||u|| n+2 Hs ). (39)
Remember now that the initial data ||u(0)||Hs is of order ε and that each term in Ms,ε has the Hs
-boundedness property. As we will see just below, the choice ψ(ε) = ε4+ρ−n will be very convenient. Using
(38), one sees that if ε is small enough then one has
|Ms,ε(u)| . ψ(ε)3+ρεn+2+ εn+2= ε2+
n
4+ρ + εn+2= o(ε2).
Integrating (39), we get ||u(t)||Hs . ε on an interval [−T, T ] such that
ε2. T ψ(ε)3+ρεn+ 1 ψ(ε) ε2+n= 2T ε2+n+4+ρn .
We can conclude that T & ε−Anholds for any fixed constant A = 1 + 1 4+ρ ∈ 1,
5 4.
6
Annex : proof of Proposition 2.4
Let us consider two numbers ℘ > 0, ℵ ≥ 0 and a sequence of numbers µj ≥ 1 such that the following two
asymptotics hold as j tends to infinity :
µj ' j℘, (40)
µj+1− µj & j−ℵ. (41)
For any m > 0 and any integer ` ∈ [0, n], we define the following two maps which look like F`
m and bFm` on [1, +∞)2n+2 (see (22)) : Hm`(ξ0, . . . , ξn+1) = ` X j=0 q ξ2 j − 1 + m2− n+1 X j=`+1 q ξ2 j − 1 + m2, b Hm`(ξ0, . . . , ξn+1) = ` X j=0 q ξ2 j − 1 + m2− n X j=`+1 q ξ2 j− 1 + m2+ q ξ2 n+1− 1 + m2. (42)
Indeed, the previous two maps are relevant to study the Klein-Gordon equation (2) with Sp(1 − ∆) = {µ2
j, j ≥ 1}. Using the asymptotic µj =
q
1 + (λ2j− 1)2∼ λ2
j and considering ℘ = 1 and ℵ ∈ (1, 2) (see
Proposition 6.1. For any ρ > 0, for almost every m > 0 (in the sense of Lebesgue), for any ` ∈ [0, n]∩N there are C > 0 and ν0> 0 such that for any (k0, . . . , kn+1) ∈ (N\{0})n+2\Ωn+2(`) we have
1 |H` m(µk0, . . . , µkn+1)| ≤ C(µk0+ µkn+1) max(ℵ ℘, 2 ℘)+ρmax(µk 1, . . . , µkn) ν0, (43) 1 |H` m(µk0, . . . , µkn+1)| ≤ C(µk0+ · · · + µkn+1) ν0. (44)
In the same spirit, if (k0, . . . , kn+1) ∈ (N\{0})n+2\Ωn+2(b`) then we have
1 | bH` m(µk0, . . . , µkn+1)| ≤ C(µk1+ · · · + µkn) ν0 µk0+ µkn+1 , (45) 1 | bH` m(µk0, . . . , µkn+1)| ≤ C(µk0+ · · · + µkn+1) ν0. (46)
In the case where ℵ belongs to (0, 1), let us consider β ∈ (0, 1] such that ℵ + (1 − β)℘ belongs to (0, 1). For any ν1> 1 there are ν0> 0 and C > 0 such that one can replace the right-hand side of (43) by
C(µk0+ µkn+1) 1+ℵ+(1−β)℘ ℘ +ρ(1 + |µβ k0− µ β kn+1|) ν1max(µ k1, . . . , µkn) ν0. (47)
Here C > 0 is independent of (k0, . . . , kn+1) and may depend on ℘, ℵ, β, n, `, m, ν1 and ρ.
Remark 6.2. In our paper, (47) is useless. We have written the proof because it requires an easy modification of that of (43) and may be useful for further developments. Let us add that (47) is indeed used in previous published works. For the Klein-Gordon equation on a torus Td, with d ≥ 4, or with a
quadratic potential on Rd, the respective spectra Sp(−∆) and Sp(−∆ + |x|2) are of the form {µ2j, j ≥ 1}, µj '
√
j and µj+1− µj & √1j. So one has ℘ = ℵ = 12. And thus, β = 1 is convenient in (47) (see [11]
and [26, line 2.3.3]).
We will follow the same idea of that of [26, Part 2.3]. Firstly, we fix the parameter ρ > 0. Secondly, since (0, +∞) is a countable union of compact intervals, it is sufficient to prove Proposition 6.1 if m belongs to a fixed compact interval J ⊂ (0, +∞) and if ` ∈ [0, n] ∩ N is also fixed. Note now that the following inequality holds
∀ξ ≥ 1 min(1, m) ≤ p
ξ2− 1 + m2
ξ ≤ max(1, m). (48) For any c ∈ R satisfying 0 < c < min 1, m,m1, the map ξ 7→
p
ξ2− 1 + m2− cξ is a positive and
increasing function on [1, +∞). Hence, the compactness of J allows us to choose a uniform constant c on the fixed compact J .
6.1
Proof of (43)
This is the big part of the proof of Proposition 6.1. Let us introduce a subset E`
J(k, α, N0) ⊂ J , for any
α > 0, N0∈ N and k = (k0, . . . , kn+1) ∈ (N\{0})n+2by the following definition :
m ∈ EJ`(k, α, N0) ⇔ |Hm`(µk0, . . . , µkn+1)| < α (µk0+ µkn+1) max(ℵ ℘, 2 ℘)+ρ(µk 1+ · · · + µkn) N0 . (49)
Since µk1+ · · · + µkn and max(µk1, . . . , µkn) are of the same order (up to a multiplicative constant
which depends on n), (43) is a consequence of the following
where we denote by Leb the Lebesgue measure on R.
We need to introduce other notations to explain the proof of (50). Let us consider another family of subsets of J for any number α > 0, σ > 0, N1∈ N and ek = (k1, . . . , kn) ∈ (N?)n :
m ∈ E0J`(ek, ασ, N1) ⇔ ∂G` m ∂m (µk1, . . . , µkn) ≤ α σ (µk1+ · · · + µkn) N1, (51)
where the map G`
mis given by G`m: [1, +∞)n → R (ξ1, . . . , ξp) 7→ ` X j=1 q m2− 1 + ξ2 j − n X j=`+1 q m2− 1 + ξ2 j,
and two families of subsets : S(ασ, N1) := k ∈ (N?)n+2\Ωn+2(`), min(µk0, µkn+1) < 1 3α2σ(µk1+ · · · + µkn) N1 , (52) Ω0n(`) := {(k1, . . . , kn) ∈ (N?)n, {k1, . . . , k`} = {k`+1, . . . , kn}}. (53)
One easily checks the following inequalities (whatever are the numbers N0, N1, σ and α)
Leb S k6∈Ωn+2(`) E` J(k, α, N0) ! ≤ Leb S k6∈Ωn+2(`) e k∈Ω0 n(`) EJ`(k, α, N0) + Leb S k6∈Ωn+2(`) e k6∈Ω0 n(`) EJ`(k, α, N0) ≤ Leb S k6∈Ωn+2(`) e k∈Ω0n(`) E` J(k, α, N0) + Leb S k∈S(ασ,N1) E` J(k, α, N0) ! Leb S k6∈Ωn+2(`) k6∈S(ασ,N1) e k6∈Ω0n(`) EJ`(k, α, N0) ∩ E0`J(ek, α σ, N 1)c + Leb S k6∈Ωn+2(`) e k6∈Ω0n(`) EJ0`(ek, ασ, N1) .
The estimates of the previous four terms are given by Lemmas 6.3,6.5,6.6 and 6.7 (see below). Indeed, it will appear that there are numbers N0, N1, δ > 0 such that for any σ ∈ (0, min(1,2NN1
0)) and α > 0 small enough we have Leb [ k6∈Ωn+2(`) EJ`(k, α, N0) ≤ 0 + Cα δ1−2σN0 N1 + Cα1−σ+ Cασδ, (54)
where C > 0 is independent of α > 0. That will obviously prove (50) by making α tend to 0+. The
Proposition 6.4 @ @ @ @ R Lemma 6.5 A A A A A A A A U @ @ @ @ R -Lemma 6.3 ? 6 -Lemma 6.7 -Lemma 6.6 (54) (50) (43)
Lemma 6.3 will show why the condition (41) is useful and hence why the Diophantine assumption r−2∈ D (see (3)) is of interest in the study of the sequence µj =p1 + (λj− 1)2.
Lemma 6.3. There is a positive constant c(J ) such that if the following three conditions hold : • 0 < α < c(J ),
• ek = (k1, . . . , kn) ∈ Ω0n(`) (see (53)),
• k06= kn+1,
then EJ`(k, α, N0) is empty.
Proof. From (41) and (48), we get |H` m(µk0, . . . , µkn+1)| = q µ2 k0− 1 + m 2−qµ2 kn+1− 1 + m 2 = |µk0− µkn+1| × |µk0+ µkn+1| |qµ2 kn+1+ 1 − m 2+qµ2 k0+ 1 − m 2| ≥ c(J ) (µk0+ µkn+1) ℵ ℘ .
From (49), the following inequality holds for any m ∈ E`
J(k, α, N0) |H` m(µk0, . . . , µkn+1)| ≤ α (µk0+ µkn+1) max(ℵ ℘,℘2)+ρ ≤ α (µk0+ µkn+1) ℵ ℘ . (55)
Choosing α small enough, we conclude that E`J(k, α, N0) is empty.
Handling the last three terms in (54) needs to introduce several notations. Let us begin with f`: [0, 1]n+3× J−1 → R (z, x0, . . . , xn+1, y) 7→ ` X j=0 q z2+ y2x2 j− n+1 X j=`+1 q z2+ y2x2 j, (56)
where we make the convention
J−1:= 1
m, m ∈ J
We also need a map g`: [0, 1]n+1× J−1 → R defined by z > 0 ⇒ g`(z, x 1, . . . , xn, y) = z ` X j=1 z q z2+ y2x2 j − n X j=`+1 z q z2+ y2x2 j , z = 0 ⇒ g`(z, x 1, . . . , xn, y) = 0.
We also define the following map ρ`
f : [0, 1]n+3→ R : ρ`f(z, x) = z if ` 6= n2, ρ`f(z, x) = z Y σ∈Sf,n X j≤n 2 (x2σ(j)− x2 j) 2 if n 2 ∈ N and ` = n 2,
where Sf,n is the set of all bijections from {0, . . . ,n2} on {n2 + 1, . . . , n + 1}. We define ρ`g replacing in
the obvious way the condition x ∈ [0, 1]n+3by x ∈ [0, 1]n+1, {0, 1, . . . , n + 1} by {1, . . . , n}, and S f,n by
Sg,n. Proposition 2.1.2 of [7] states the following result.
Proposition 6.4. There are numbers eN ∈ N, α0> 0, δ > 0, C > 0 such that for any ` ∈ [0, n + 1] ∩ N,
any α ∈ (0, α0), N ≥ eN and (z, x) ∈ [0, 1]n+3with ρ`f(z, x) > 0, the Lebesgue measure of the subset
I`f(z, x, α) := {y ∈ J−1, |f`(z, x, y)| < αρ`f(z, x) N}
is less than Cαδρ`
f(z, x)N δ. The same is true by replacing respectively f, [0, 1]n+3, ρ`f by g, [0, 1]n+1, ρ`g.
Furthermore, there is K = K(N ) ∈ N such that the set I`g(z, x, α) may be written as the union of at most
K open disjoints subintervals of J−1.
The following lemma gives an upper bound of the Lebesgue measures of the subsets E0`
J(ek, ασ, N1).
Lemma 6.5. Under the assumptions of Proposition 6.4, there are constants C1> 0, M ∈ N? such that
for any α > 0 and σ > 0 with ασ∈ (0, α
0), N1∈ N with N1> max(M eN ,nMδ℘ ), one has
Leb [ k6∈Ωn+2(`) e k6∈Ω0n(`) EJ0`(ek, ασ, N1) ≤ C1ασδ.
Furthermore, each subset E0`
J(ek, ασ, N1) may be written as at most K = K(N1) disjoints subintervals of
J .
Proof. The proof will be a consequence of Proposition 6.4. Let us introduce
X := (z, x) ∈ [0, 1]n+1, ∃ek ∈ (N?)n z = n X j=1 µkj −1 , ∀j ∈ [1, n] ∩ N xj= z q µ2 kj− 1 .
We also use the subset X0n
` ⊂ X of elements (z, x) which correspond to an element ek ∈ Ω0n(`) according
to the definition of X. Note now that one has obviously ρ`g(z, x) ≤ C(n)z for any (z, x) ∈ X. Let us explain why we can say more if (z, x) belongs to X\X0n
` . Firstly, one easily checks that ρ`g(z, x) is not
zero. It is indeed clear if ` 6= n2 because ρ`
g(z, x) = z > 0 (see the definition of X). If n is even and
if ` equals n2, then there are two integers j ∈ [1,n2] and j0 ∈ [n
2, n] such that kj 6= kj0. Consequently,
µkj and µkj0 are different and the definition (57) forces ρ
`
g(z, x) to be positive. Secondly, by using the
“Diophantine condition” (41) we have
In other words, there is a positive constant M > 0 such that zM ≤ ρ`
g(z, x) ≤ Cz holds provided that ek
does not belong to Ω0n(`). We can go on as in [26, Lemma 2.3.3] by noticing
∂G` m ∂m (µk1, . . . , µkn) = ` X j=1 m q m2+ µ2 kj − 1 − n X j=`+1 m q m2+ µ2 kj − 1 = 1 zg`(z, x1, . . . , xn, y),
where y is nothing else than m1. Hence, if m belongs to E0`
J(ek, ασ, N1) then one has
|g`(z, x
1, . . . , xn, y)| < ασzN1+1≤ ασ[ρ`g(z, x)]
N1+1 M .
We now combine Proposition 6.4 and the fact that m ∈ J 7→ m−1∈ J−1is a bi-Lipschitz diffeomorphism,
we get Leb hEJ0`(ek, ασ, N1) i ≤ C(J )ασδz(N1+1M )δ= Cα σδ (µk1+ · · · + µkn) δN1+1M .
Summing in ek = (k1, . . . , kn) and using the asymptotic µk ' k℘(see (40)), we get the conclusion.
To bound the Lebesgue measures of the subsets E`
J(k, α, N0), we have to make use of the sets S(ασ, N1)
introduced in (52).
Lemma 6.6. Under the assumptions of Proposition 6.4, there are constants M, C > 0 and θ ∈ (0, 1) such that for any (N0, N1) ∈ (N?)2 satisfying N0 > max
e N M N1, (n + 2)M Nδ℘1 , any (α, σ) ∈ (0, +∞)2 with α + ασ+ α1−2σN0N1 < θ one has Leb [ k∈S(ασ,N 1) EJ`(k, α, N0) ≤ C2α δ1−2σN0 N1 . Proof.
As in Lemma 6.5, it is sufficient to get an adequate bound of LebEJ`(k, α, N0) with k ∈ S(ασ, N1).
We consider two cases.
First case. If µk0 + µkn+1 >
1
α2σ(µk1 + · · · + µkn)
N1 then the definition (52) implies the following
bound from below :
max µk0, µkn+1 ≥
2
3α2σ(µk1+ · · · + µkn)
N1.
Suppose for instance that max µk0, µkn+1 = µk0holds. Then we have µkn+1 <
1 3α
−2σ(µ
k1+· · ·+µkn)
N1.
By remembering that N1 and the real numbers µj are greater than or equal to 1, we can find c(J ) ≥ 1
which depends only on the fixed compact J (see the discussion after (48)) such that
Hm`(µk0, . . . , µkn+1) ≥ q µ2 k0− 1 + m 2−qµ2 kn+1− 1 + m 2− n X j=1 q µ2 kj − 1 + m 2 ≥ 1 c(J ) µk0− µkn+1 − c(J) n X j=1 µkj ≥ 1 3c(J )α2σ − c(J ) (µk1+ · · · + µkn) N1. (57)
If ασ is small enough, we get H`
m(µk0, . . . , µkn+1) ≥ 1. If α < 1 moreover holds, then the set E
`
J(k, α, N0)
is empty because |H`
m(µk0, . . . , µkn+1)| ≤ α holds for any m ∈ E
`
J(k, α, N0) (see (49)).
Second case. We assume that µk0+ µkn+1 ≤ α
−2σ(µ
k1+ · · · + µkn)
N1 holds. Since ασbelongs to (0, 1),
we get
µk0+ µk1+ · · · + µkn+ µkn+1 ≤ 2α
−2σ(µ
k1+ · · · + µkn)
That ensures that we have for any m ∈ EJ`(k, α, N0) |H` m(µk0, . . . , µkn+1)| ≤ α (µk1+ · · · + µkn) N0 ≤ 2N0N1α1−2σN0N1 (µk0+ · · · + µkn+1) N0/N1. (58)
We then note that
f` 1 µk0+ · · · + µkn+1 , q µ2 k0− 1 µk0+ · · · + µkn+1 , . . . , q µ2 kn+1 − 1 µk0+ · · · + µkn+1 , 1 m
is nothing else than m(µ 1
k0+···+µkn+1)H
m
` (µk0, . . . , µkn+1) and so is less than or equal to
2N0N1 inf(J )α 1−2σN0N1 1 (µk0+ · · · + µkn+1) 1+N0 N1 .
We can conclude by using the same strategy we used in the proof of Lemma 6.5 but with f` instead of g` (defined in (56)). Let us give a sketch of the proof. Using the fact that k does not belong to Ωn+2(`)
allows for the following bound
f` 1 µk0+···+µkn+1, q µ2 k0−1 µk0+···+µkn+1, . . . , q µ2 kn+1−1 µk0+···+µkn+1, 1 m ! < 2 N0 N1 inf(J )α 1−2σN0 N1ρ` f 1 µk0+···+µkn+1, q µ2 k0−1 µk0+···+µkn+1, . . . , q µ2 kn+1−1 µk0+···+µkn+1 !M1 1+N0N1 ,
for some constant M > 0. An application of Proposition 6.4 gives the conclusion.
Lemma 6.7. Under the assumptions of Lemma 6.5, if N0 > N1+n℘ holds and if ασ is small enough
then one can find C3> 0 such that
