Sergei Kuksin

Asymptotic properties of integrals of quotients, when the numerator oscillates and denominator degenerate

Keywords: asymptotic of integrals, oscillating integrals, four-waves interaction Mathematical Subject Classification 2010: 34E05, 34E10

Asymptotic properties of integrals of quotients, when the numerator oscillates and denominator degenerate

Sergei Kuksin

Introduction

In the paper [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF] we study asymptotical, as ν → 0, behaviour of integrals

I ν = R d ×R d dx dy F (x, y) (x • y) 2 + (νΓ(x, y)) 2 , d ≥ 2 , 0 < ν ≤ 1,
where F and Γ are C 2 -function, Γ is positive and the two satisfy certain conditions at infinity. In particular, if Γ ≡ 1, then

|∂ α z F (z)| ≤ C z -N -|α| ∀z = (x, y) ∈ R 2d , ∀ |α| ≤ 2 , (1.1)
where C > 0 and N > 2d -2. Denote by

Σ ⊂ R 2d = R d x × R d y (1.2)
the quadric {(x, y) : x • y = 0}, and by Σ * its regular part Σ \ {(0, 0)}. It is proved in [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF] (see [START_REF] Yu | One approach to the computation of asymptotics of integrals of rapidly varying functions[END_REF] for related results) that

I ν = πν -1 Σ * F (z) |z| Γ(z) d Σ * z + O(χ d (ν)), (1.3)
where

χ d (ν) = 1, d ≥ 3 , max ln(ν -1 ), 1 , d = 2 , c Sergei Kuksin, 20XX
d Σ * is the volume element on Σ * , induced from the standard Riemann structure in R 2d , and the integral in (1.3) converges absolutely. Integrals of this form appear in the study of the four-waves interaction. The wave turbulence (WT) limit in systems with the four-waves interaction leads to oscillating versions of the integrals above with constant functions Γ. Re-denoting νΓ back to ν we write the integrals in question as

J ν = R 2d dz F (z) cos(λx • y) (x • y) 2 + ν 2 , d ≥ 2 , λ ≥ 0 , 0 < ν ≤ 1 (1.4)
(as before, z = (x, y)). We assume that F is a C 2 -function, satisfying (1.1). The aim of this work is to prove the following result, describing the asymptotical behaviour of J ν when ν → 0, uniformly in λ ≥ 0: Theorem 1.1. Let 0 < ν ≤ 1 and λ ≥ 0. Then the integral J ν and the integral

J 0 = πe -νλ Σ * F (z)|z| -1 d Σ * z converge absolutely and J ν -ν -1 J 0 ≤ Cχ d (ν), (1.5)
where C depends on d and the constants C , N in (1.1), but not on ν and λ.

Note that since C does not depend on λ, then relation (1.5) remains valid for integrals (1.4), where λ = λ(ν) is any function of ν. Concerning the imposed restriction d ≥ 2 see item iv) in Section 5.

If λ = 0, the integral J ν becomes a special case of I ν (with Γ = 1), and (1.5) follows from (1.3). Since sin 2 ( λ 2 x • y) = 1 2 (1 -cos(λx • y)), then combining (1.3) and (1.5) we get Corollary 1.2. As ν → 0,

R d ×R d dx dy F (x, y) sin 2 ( λ 2 x • y) (x • y) 2 + ν 2 = π 2 ν -1 (1 -e -νλ ) Σ * F (z) |z| d Σ * z + O(χ d (ν)) , (1.6)
uniformly in λ ≥ 0.

Classically the WT considers singular versions of the integral in the l.h.s. of (1.6):

dx dy F (x, y) sin 2 ( λ 2 x • y) (x • y) 2 .
(1.7)

The theory deals with these integrals by performing certain formal calculations, see Section 6 of [START_REF] Nazarenko | Wave Turbulence[END_REF] (e.g. note there eq. (6.39)+(6.41)). Assertion (1.6) may be regarded as a regularisation of the integral (1.7). The factor |z| -1 which it introduces in the limiting density is not present in the asymptotical description of integrals (1.7), used in the works on WT.

Theorem 1.1 is proved below in Sections 2-4, using the geometric approach of the paper [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF], which also applies to various modifications of integrals I ν and J ν . Some of these applications are discussed in the last Section 5.

Notation. As usual, we denote z = |z| 2 + 1. For an integral

I = R 2d f (z) dz and a submanifold M ⊂ R 2d , dim M = m ≤ 2d, compact or not (if m = 2d, then M is an open domain in R 2d ) we write I, M = M f (z) d M (z), where d M (z) is the volume-element on M , induced from R 2d . Similar |I|, M stands for the integral M |f (z)| d M (z).
2. Geometry of the quadric {x • y = 0} and its vicinity.

2.1. The geometry of the quadric. The difficulty in studying the integral J ν with small ν comes from the vicinity of the quadric Σ = {x • y = 0}. To examine the integral's behaviour there we first analyse the geometry of the vicinity of the regular part of the quadric Σ * = Σ \ {(0, 0)}, following [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF]. Example 5.1 at the end of the paper provides an elementary illustration to the objects, involved in this analysis.

The set Σ * is a smooth submanifold of R 2d of dimension 2d -1. We denote by ξ a local coordinate on Σ * with the coordinate mapping ξ

→ (x ξ , y ξ ) = z ξ ∈ Σ * , denote |ξ| = |(x ξ , y ξ )| and denote N (ξ) = (y ξ , x ξ ). The latter is the normal to Σ * at ξ of length |ξ|. For any 0 ≤ R 1 < R 2 we set S R 1 = {z ∈ R 2d : |z| = R 1 } , Σ R 1 = Σ ∩ S R 1 , S R 2 R 1 = {z : R 1 < |z| < R 2 } , Σ R 2 R 1 = Σ ∩ S R 2 R 1 ,
and for t > 0 denote by D t the dilation operator

D t : R 2d → R 2d , z → tz .
For z = (x, y) we write ω(z) = x • y.

The following result is Lemma 3.1 from [1]:

Lemma 2.1. 1) There exists θ 0 ∈ (0, 1] such that a suitable neighbourhood

Σ nbh = Σ nbh (θ 0 ) of Σ * in R 2d \ {0}
, is invariant with respect to the dilations D t , t > 0, and may be uniquely parametrized as

Σ nbh = {π(ξ, θ) : ξ ∈ Σ * , |θ| < θ 0 } , where π(ξ, θ) = (x ξ , y ξ ) + θN ξ = (x ξ , y ξ ) + θ(y ξ , x ξ ). In particular, |π(ξ, θ)| 2 = |ξ| 2 (1 + θ 2 ). 2) If π(ξ, θ) ∈ Σ nbh , then ω π(ξ, θ) = |ξ| 2 θ. (2.1) 3) If (x, y) ∈ S R \ Σ nbh , then |x • y| ≥ cR 2 for some c = c(θ 0 ) > 0. For 0 ≤ R 1 < R 2 we will denote (Σ nbh ) R 2 R 1 = π(Σ R 2 R 1 × -θ 0 , θ 0 ) .
Now we discuss the Riemann geometry of the domain Σ nbh = Σ nbh (θ 0 ), following [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF]. The set Σ is a cone with the vertex in the origin, and Σ * = {tz : t > 0, z ∈ Σ 1 }. The set Σ 1 is a closed manifold of dimension 2d -2. Let us cover it by a finite system of charts N 1 , . . . , N ñ, N j = {η j = (η j 1 , . . . , η j 2d-2 )}, and for any chart N j denote by m(dη j ) the volume element on Σ 1 , induced from R 2d . Below we write points in any chart N j as η, and the volume element -as m(dη).

The mapping

Σ 1 × R + → Σ * , ((x η , y η ), t) → D t (x η , y η )
is a diffeomorphism. Accordingly, we can cover Σ * by the ñ charts N j × R + with the coordinates (η j , t) =: (η, t). The coordinates (η, t, θ), where η ∈ N j , t > 0 and |θ| < θ 0 , 1 ≤ j ≤ ñ, make coordinate systems on Σ nbh = Σ nbh (θ 0 ). In the coordinates (η, t) the volume element on Σ * is

d Σ * = t 2d-2 m(dη) dt . (2.2)
In the coordinates (η, t, θ) the volume elements in R 2d reeds dx dy = t 2d-1 µ(η, θ)m(dη)dt dθ , where µ(η, 0) =

(see [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF]), a dilation map D r , r > 0, reeds D r (η, t, θ) = (η, rt, θ) , and by (2.1)

ω(η, t, θ) = t 2 θ . (2.4)
Finally, since at a point z = π(ξ, θ) ∈ Σ nbh we have ∂ ∂θ = ∇ z • (y ξ , x ξ ), then in view of (1.1) for any (η, t, θ) and any k ≤ 2,

∂ k ∂θ k F (η, t, θ) ≤ C t -N , N > 2d -4.
(2.5)

2.2. The volume element d Σ * and the measure |z| -1 d Σ * . Theorem 1.1 and the result of [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF] (see (1.3)) show that the manifold Σ * , equipped with the measure |z| -1 d Σ * , is crucial to study asymptotic of integrals I ν , J ν and their similarities (cf. Section 6 of [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF] and Section 5 below). The coordinates (η, t) and the presentation (2.2) for the volume element are sufficient for the purposes of this work. But the quadric Σ is reach in structures and admits more instrumental coordinate systems. In particular, if d = 2 we can introduce in the space R 2 x in (1.2) the polar coordinates (r, ϕ). Then for any fixed non-zero vector x = (r, ϕ) ∈ R 2

x the set {y ∈ R 2 y : (x, y) ∈ Σ * } is the line in R 2 y , perpendicular to x, and having the angle ϕ + π/2 with the horizontal axis. Parametrizing it by the length-coordinate l we get on Σ * the coordinates (r, l, ϕ) ∈ R + × R × S 1 , S 1 = R/2πZ, with the coordinate mapping Φ : (r, l, ϕ) → x = (r cos ϕ, r sin ϕ), y = (-l sin ϕ, l cos ϕ) (this map is singular at r = 0). Since |∂Φ/∂r| 2 = 1, |∂Φ/∂l| 2 = 1, |∂Φ/∂ϕ| 2 = r 2 + l 2 , ∂Φ/∂r, ∂Φ/∂l = ∂Φ/∂r, ∂Φ/∂ϕ = ∂Φ/∂l, ∂Φ/∂ϕ = 0, then in these coordinates the volume element on Σ * reeds as √ r 2 + l 2 drdldϕ, and the measure |z| -1 d Σ * -as dr dl dϕ. Consider the fibering

Π : R 2 x × R 2 y ⊃ Σ * → R 2 x , (x, y) → x. It has a singular fiber Π -1 0 = {0} × R 2
y , and for any non-zero x the fiber Π -1 x equals {x} × x ⊥ , where x ⊥ is the line in R 2 y , perpendicular to x. Since dx = rdr dϕ, then the given above presentation for the measure |z| -1 d Σ * implies that its restriction to the regular part Σ + * of the fibered manifold Σ * , Σ + * = Σ * \ ({0} × R 2 y ), disintegrates by the foliation Π as

(|z| -1 d Σ * ) | Σ + * = |x| -1 dx d x ⊥ y, x = 0, y ∈ x ⊥ , (2.6)
where d x ⊥ is the length on the euclidean line x ⊥ ⊂ R 2 y . We do not undertake the job of getting a right analogy of this result for the multidimensional case d > 2, but note that a straightforward modification of the construction above leads to the observation that for any d ≥ 2 the measure |z| -1 d Σ * , restricted to Σ + * , disintegrates as

p d (x, y)dx d x ⊥ y, x ∈ R d \ {0}, y ∈ x ⊥ , (2.7)
where x ⊥ is the orthogonal complement to x in R d y , , d x ⊥ is the volume element on this euclidean space, and the function p d satisfies the estimate

p d ≤ C(|x| + |y|) d-2 |x| 1-d .

Integral over the vicinity of Σ

To study the behaviour of the integral over a neighbourhood of Σ we first prove that the integral, evaluated over the vicinity of the singular point (0, 0) is small, and next study the integral over the vicinity of the regular part Σ * of the quadric.

For 0 < δ ≤ 1 denote

K δ = {z = (x, y) : |x| ≤ δ, |y| ≤ δ} ⊂ R d × R d .
An upper bound for the integral over K δ follows from Lemma 2.1 of [START_REF] Kuksin | Asymptotical expansions for some integrals of quotients with degenerated divisors[END_REF]:

| |J ν |, K δ | ≤ K δ |F (z)| dz (x • y) 2 + ν 2 ≤ Cν -1 δ 2d-2 . ( 3.1)
Now we estimate the integral over the neighbourhood Σ nbh of Σ * . For this end, using (2.3), for 0

≤ A < B ≤ ∞ we disintegrate J ν , (Σ nbh ) B A as J ν , (Σ nbh ) B A = Σ 1 m(dη) B A dt t 2d-1 θ 0 -θ 0 dθ F (η, t, θ)µ(η, θ) cos(λx • y) t 4 θ 2 + ν 2 = Σ 1 m(dη) B A dt t 2d-1 Υ ν (η, t) , (3.2)
where

Υ ν (η, t) = t -4 θ 0 -θ 0 F (η, t, θ)µ(η, θ) cos(λt 2 θ) dθ θ 2 + ε 2 , ε = νt -2 .
To study Υ ν we first consider the integral Υ 0 ν , obtained from Υ ν by frozening Fµ at θ = 0. Since µ(η, 0) = 1, then

Υ 0 ν = 2t -4 F (η, t, 0) θ 0 0 cos(λt 2 θ) d θ θ 2 + ε 2 = 2ν -1 t -2 F (η, t, 0) θ 0 /ε 0 cos(νλw) dw w 2 + 1 .
Consider the integral 2

θ 0 /ε 0 cos(νλw) dw w 2 + 1 = 2 ∞ 0 cos(νλw) dw w 2 + 1 -2 ∞ θ 0 /ε cos(νλw) dw w 2 + 1 =: I 1 -I 2 . Since 2 ∞ 0 cos(ξw) dw w 2 + 1 = ∞ -∞ e iξw dw w 2 + 1 = πe -|ξ| ,
then I 1 = πe -νλ . For I 2 we have an obvious bound

|I 2 | ≤ 2ε/θ 0 = C 1 νt -2 . So Υ 0 ν (η, t) = πν -1 t -2 F (η, t, 0)(e -νλ + ∆ t ) , |∆ t | ≤ Cνt -2 . (3.3)
Now we estimate the difference between Υ ν and Υ 0 ν . Writing (F µ)(η, t, θ) -(F µ)(η, t, 0) as A(η, t)θ + B(η, t, θ)θ 2 , where |A|, |B| ≤ C t -N in view of (2.5), we have

Υ ν -Υ 0 ν = t -4 θ 0 -θ 0 (Aθ + Bθ 2 ) cos(λt 2 θ) dθ θ 2 + ε 2 .
Since the first integrand is odd in θ, then its integral vanishes, and

|Υ ν -Υ 0 ν | ≤ C t -N t -4 θ 0 -θ 0 θ 2 dθ θ 2 + ε 2 ≤ 2C t -N t -4 θ 0 .
So by (3.3)

|Υ ν (η, t) -πν -1 t -2 F (η, t, 0)e -νλ | ≤ C t -N t -4 + ν -1 t -2 νt -2 ≤ C t -N t -4 . (3.4)
4. End of the proof of Theorem 1.1

1) In view of (3.2), (3.4) and since N > 2d -2, for δ ∈ (0, 1] we have

J ν , Σ nbh ∞ δ -πν -1 e -νλ Σ 1 m dη ∞ δ dt t 2d-3 F (η, t, 0) ≤ C ∞ δ t 2d-5 t -N dt ≤ C 1 χ d (δ) .
2) Since d ≥ 2 and N > 2d -2, then by estimate (2.5) the integral Σ 1 m dη ∞ 0 dt t 2d-3 F (η, t, 0) converges absolutely, and by (2.2) it equals

Σ 1 m dη ∞ 0 dt t 2d-3 F (η, t, 0) = Σ * |z| -1 F (z) d Σ * z .
3) Applying 1) and 2) to F replaced by F 0 = C z -N and using that |F | ≤ |F 0 | by (1.1) we find that the integral J ν , Σ nbh ∞ δ also converges absolutely.

4) As

|π(ξ, θ)| ≤ √ 2 |ξ|, then (Σ nbh ) δ 0 ⊂ S √ 2δ 0 ⊂ K √ 2δ . Therefore by (3.1) J ν , Σ nbh δ 0 -πν -1 e -νλ Σ 1 m dη δ 0 dt t 2d-3 F (η, t, 0) ≤ |J ν |, K √ 2δ + πν -1 e -νλ Σ 1 m dη δ 0 dt t 2d-3 |F (η, t, 0)| ≤ C 1 ν -1 δ 2d-2 + C 2 ν -1 δ 2 ,
for any 0 < δ ≤ 1. Choosing δ = √ ν, from here and 1)-3) we find that So the integral J ν converges absolutely and, in view of 4) and 2),

J ν , Σ nbh -πν -1 e -νλ
J ν -πν -1 e -νλ

Comments

i) The only part of the proof, where we use that N > 2d -2 is Step 2) in Section 4: there this relation is evoked to establish the absolute convergence of the integral J 0 ; everywhere else it suffices to assume that N > 2d -4. Accordingly,

dt t 2d- 3 Fν dr r 2d- 1 S 2 + ν 2 .

 3122 (η, t, 0) ≤ Cχ d (ν), and that the integral J ν , Σ nbh converges absolutely. 5) Finally, let us estimate the integral over R 2d \ Σ nbh :|J ν |, R 2d \ Σ nbh ≤ {|z|≤ √ ν} |F | dz ω 2 + ν 2 + C d ∞ √ r \Σ nbh |F (z)| d S r ωBy item 3) of Lemma 2.1, |ω| ≥ Cr 2 in S r \ Σ nbh . Jointly with (3.1) this implies that|J ν |, R 2d \ Σ nbh ≤ C + C ∞ √ νr 2d-1 r -4 r -N dr ≤ C 1 χ d (ν).

dt t 2d- 3 F|z| - 1 F

 31 (η, t, 0) = J ν -πν -1 e -νλ Σ * (z) d Σ * z ≤ Cχ d (ν) .

(4. 1 )

 1 This proves Theorem 1.1.

if F satisfies (1.1) with N > 2d -4 and |F |, Σ ∞ 1 < ∞, then (1.5) holds, since |F |, Σ 1 0 < ∞, see Step 4) Section 4. ii) Our approach does not apply to study integrals (1.4), where the divisor (x • y) 2 + ν 2 is replaced by (x • y) 2 + (νΓ(x, y)) 2 and Γ = Const. But it applies to integrals

under certain restrictions on λ. E.g., if 1 ≤ λ ≤ ν -1 and d ≥ 3, then J s ν = O(1) as ν → 0, and the leading term again is given by an integral over Σ * . The case d = 2 is a bit more complicated.

iii) The approach allows to study integrals (1.4), where the quadratic form z → x • y is replaced by any non-degenerate indefinite quadratic form of z ∈ R M , M ≥ 4.

iv) The restriction M ≥ 4 in iii) (and d ≥ 2 in the main text, where dim z = 2d) was imposed since near the origin the disparity (4.1) is controlled by the integral 0 t M -5 dt, which strongly diverges if M < 4. The difficulty disappears if F vanishes near zero. This may be illustrated by the following easy example:

Example 5.1. Consider

where F ∈ C 2 0 (R 2 ) vanishes near the origin. Now 2d = 2, the quadric Σ = {xy = 0} is one dimensional, has a singularity at the origin and its smooth part Σ * = Σ \ 0 has four connected components. Consider one of them: C 1 = {(x, y) : y = 0, x > 0}. Now the coordinate ξ is a point in R + with (x ξ , y ξ ) = (ξ, 0) and with the normal N (ξ) = (0, ξ), the set Σ 1 ∩ C 1 is the single point (1, 0) and the coordinate (η, t, θ) in the vicinity of C 1 degenerates to (t, θ), t > 0, |θ| < θ 0 , with the coordinate-map (t, θ) → (t, tθ). The relations (2.2) and (2.3) are now obvious, and the integral (3.1) vanishes if δ > 0 is sufficiently small. Interpreting z = (x, y) as a complex number, we write the assertion of Theorem 1.1 as

where the integral is a contour integral in the complex plane.

Supports. We acknowledge the support from the Centre National de la Recherche Scientifique (France) through the grant PRC CNRS/RFBR 2017-2019 No 1556, and from the Russian Science Foundation through the project 18-11-00032.