Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates
We study asymptotic expansion as ν→0 for integrals over ℝ²d={(x,y)} of quotients of the form F(x,y)cos(λx∙y)/((x∙y)²+ν²), where λ≥0 and F decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
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irk-123456789-1458832019-02-03T01:23:14Z Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates Kuksin, S. We study asymptotic expansion as ν→0 for integrals over ℝ²d={(x,y)} of quotients of the form F(x,y)cos(λx∙y)/((x∙y)²+ν²), where λ≥0 and F decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence. Ми вивчаємо асимптотичне поводження при ν→0 iнтегралiв в ℝ²d = {(x,y)} вiд виразiв вигляду F(x,y)cos(λx∙y)/((x∙y)²+ν²), де λ≥0 i F досить швидко спадає на нескiнченностi. Подiбнi iнтеграли виникають в теорi ї хвильової турбулентностi. 2018 Article Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates / S. Kuksin // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 510-518. — Бібліогр.: 3 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.04.510 Mathematics Subject Classification 2000: 34E05, 34E10 http://dspace.nbuv.gov.ua/handle/123456789/145883 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study asymptotic expansion as ν→0 for integrals over ℝ²d={(x,y)} of quotients of the form F(x,y)cos(λx∙y)/((x∙y)²+ν²), where λ≥0 and F decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence. |
| format |
Article |
| author |
Kuksin, S. |
| spellingShingle |
Kuksin, S. Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates Журнал математической физики, анализа, геометрии |
| author_facet |
Kuksin, S. |
| author_sort |
Kuksin, S. |
| title |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates |
| title_short |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates |
| title_full |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates |
| title_fullStr |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates |
| title_full_unstemmed |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates |
| title_sort |
asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/145883 |
| citation_txt |
Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates / S. Kuksin // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 510-518. — Бібліогр.: 3 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT kuksins asymptoticpropertiesofintegralsofquotientswhenthenumeratoroscillatesandthedenominatordegenerates |
| first_indexed |
2025-07-10T22:48:03Z |
| last_indexed |
2025-07-10T22:48:03Z |
| _version_ |
1837301959894237184 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 4, pp. 510–518
doi: https://doi.org/10.15407/mag14.04.510
Asymptotic Properties of Integrals of
Quotients when the Numerator Oscillates
and the Denominator Degenerates
Sergei Kuksin
Dedicated to V.A. Marchenko on the occasion of his 95th birthday
We study asymptotic expansion as ν → 0 for integrals over R2d = {(x, y)}
of quotients of the form F (x, y) cos(λx · y)
/(
(x · y)2 + ν2
)
, where λ ≥ 0 and
F decays at infinity sufficiently fast. Integrals of this kind appear in the
theory of wave turbulence.
Key words: asymptotic of integrals, oscillating integrals, four-waves in-
teraction.
Mathematical Subject Classification 2010: 34E05, 34E10.
1. Introduction
In the paper [2] we study asymptotic behaviour of integrals
Iν =
∫
Rd×Rd
dx dy
F (x, y)
(x · y)2 + (νΓ(x, y))2
, d ≥ 2, 0 < ν ≤ 1,
as ν → 0, where F and Γ are C2-functions, Γ is positive and the two satisfy
certain conditions at infinity. In particular, if Γ ≡ 1, then
|∂αz F (z)| ≤ C ′ 〈z〉−N−|α| , z = (x, y) ∈ R2d, |α| ≤ 2, (1.1)
where C ′ > 0 and N > 2d− 2. Denote by
Σ ⊂ R2d = Rdx × Rdy (1.2)
the quadric {(x, y) : x ·y = 0}, and by Σ∗ its regular part Σ\{(0, 0)}. It is proved
in [2] (see [1] for related results) that
Iν = πν−1
∫
Σ∗
F (z)
|z|Γ(z)
dΣ∗z +O(χd(ν)), (1.3)
where
χd(ν) =
{
1, d ≥ 3,
max
(
ln(ν−1), 1
)
, d = 2,
c© Sergei Kuksin, 2018
https://doi.org/10.15407/mag14.04.510
Asymptotic Properties of Integrals of Quotients. . . 511
dΣ∗ is the volume element on Σ∗, induced from the standard Riemann structure
in R2d, 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
Jν =
∫
R2d
dz
F (z) cos(λx · y)
(x · y)2 + ν2
, d ≥ 2, λ ≥ 0, 0 < ν ≤ 1 (1.4)
(as before, z = (x, y)). We assume that F is a C2-function, satisfying (1.1).
The aim of this work is to prove the following result, describing the asymptotic
behaviour of Jν when ν → 0, uniformly in λ ≥ 0:
Theorem 1.1. Let 0 < ν ≤ 1 and λ ≥ 0. Then the integral Jν and the
integral
J0 = πe−νλ
∫
Σ∗
F (z)|z|−1 dΣ∗z
converge absolutely and ∣∣Jν − ν−1J0
∣∣ ≤ Cχd(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 Jν becomes a special case of Iν (with Γ = 1), and (1.5)
follows from (1.3). Since sin2(λ2x · y) = 1
2(1 − cos(λx · y)), then combining (1.3)
and (1.5) we get
Corollary 1.2. As ν → 0,∫
Rd×Rd
dx dy
F (x, y) sin2(λ2x · y)
(x · y)2 + ν2
=
π
2
ν−1(1− e−νλ)
∫
Σ∗
F (z)
|z|
dΣ∗z +O(χd(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) sin2(λ2x · y)
(x · y)2
. (1.7)
The theory deals with these integrals by performing certain formal calculations,
see Section 6 of [3] (e.g., note there equations (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 asymptotic 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 [2], which also applies to various modifications of integrals Iν and
Jν . Some of these applications are discussed in the last Section 5.
512 Sergei Kuksin
Notation. As usual, we denote 〈z〉 =
√
|z|2 + 1. For an integral
I =
∫
R2d
f(z) dz
and a submanifold M ⊂ R2d, dimM = m ≤ 2d, compact or not (if m = 2d, then
M is an open domain in R2d) we write
〈I,M〉 =
∫
M
f(z) dM (z),
where dM (z) is the volume-element on M , induced from R2d. Similar 〈|I|,M〉
stands for the integral
∫
M |f(z)| dM (z).
2. Geometry of the quadric {x · y = 0} and its vicinity
2.1. The geometry of the quadric. The difficulty in studying the in-
tegral Jν 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 [2]. Exam-
ple 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 R2d of dimension 2d − 1. We denote
by ξ a local coordinate on Σ∗ with the coordinate mapping ξ 7→ (xξ, yξ) = zξ ∈
Σ∗, denote |ξ| = |(xξ, yξ)| and denote N(ξ) = (yξ, xξ). The latter is the normal
to Σ∗ at ξ of length |ξ|. For any 0 ≤ R1 < R2 we set
SR1 = {z ∈ R2d : |z| = R1}, ΣR1 = Σ ∩ SR1 ,
SR2
R1
= {z : R1 < |z| < R2}, ΣR2
R1
= Σ ∩ SR2
R1
,
and for t > 0 denote by Dt the dilation operator
Dt : R2d → R2d, z 7→ tz.
For z = (x, y) we write ω(z) = x · y.
The following result is Lemma 3.1 from [2]:
Lemma 2.1.
1) There exists θ0 ∈ (0, 1] such that a suitable neighbourhood Σnbh = Σnbh(θ0)
of Σ∗ in R2d \ {0}, is invariant with respect to the dilations Dt, t > 0, and
may be uniquely parametrized as
Σnbh = {π(ξ, θ) : ξ ∈ Σ∗, |θ| < θ0},
where π(ξ, θ) = (xξ, yξ)+θNξ = (xξ, yξ)+θ(yξ, xξ). In particular, |π(ξ, θ)|2 =
|ξ|2(1 + θ2).
2) If π(ξ, θ) ∈ Σnbh, then
ω
(
π(ξ, θ)
)
= |ξ|2θ. (2.1)
Asymptotic Properties of Integrals of Quotients. . . 513
3) If (x, y) ∈ SR \ Σnbh, then |x · y| ≥ cR2 for some c = c(θ0) > 0.
For 0 ≤ R1 < R2 we will denote
(Σnbh)R2
R1
= π(ΣR2
R1
×
(
− θ0, θ0)
)
.
Now we discuss the Riemann geometry of the domain Σnbh = Σnbh(θ0), follow-
ing [2].
The set Σ is a cone with the vertex in the origin, and Σ∗ = {tz : t > 0, z ∈
Σ1}. The set Σ1 is a closed manifold of dimension 2d − 2. Let us cover it by
a finite system of charts N1, . . . ,Nñ, Nj = {ηj = (ηj1, . . . , η
j
2d−2)}, and for any
chart Nj denote by m(dηj) the volume element on Σ1, induced from R2d. Below
we write points in any chart Nj as η, and the volume element — as m(dη).
The mapping
Σ1 × R+ → Σ∗, ((xη, yη), t)→ Dt(xη, yη)
is a diffeomorphism. Accordingly, we can cover Σ∗ by the ñ charts Nj ×R+ with
the coordinates (ηj , t) =: (η, t). The coordinates (η, t, θ), where η ∈ Nj , t > 0
and |θ| < θ0, 1 ≤ j ≤ ñ, make coordinate systems on Σnbh = Σnbh(θ0). In the
coordinates (η, t) the volume element on Σ∗ is
dΣ∗ = t2d−2m (dη) dt. (2.2)
In the coordinates (η, t, θ) the volume elements in R2d reeds
dx dy = t2d−1µ(η, θ)m (dη) dt dθ, where µ(η, 0) = 1 (2.3)
(see [2]), a dilation map Dr, r > 0, reeds Dr(η, t, θ) = (η, rt, θ), and by (2.1)
ω(η, t, θ) = t2θ. (2.4)
Finally, since at a point z = π(ξ, θ) ∈ Σnbh we have ∂
∂θ = ∇z · (yξ, xξ), then in
view of (1.1) for any (η, t, θ) and any k ≤ 2,∣∣∣∣ ∂k∂θkF (η, t, θ)
∣∣∣∣ ≤ C 〈t〉−N , N > 2d− 4. (2.5)
2.2. The volume element dΣ∗ and the measure |z|−1dΣ∗. Theorem 1.1
and the result of [2] (see (1.3)) show that the manifold Σ∗, equipped with the
measure |z|−1dΣ∗ , is crucial to study asymptotic of integrals Iν , Jν and their
similarities (cf. Section 6 of [2] 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 R2
x
in (1.2) the polar coordinates (r, ϕ). Then for any fixed non-zero vector x =
(r, ϕ) ∈ R2
x the set {y ∈ R2
y : (x, y) ∈ Σ∗} is the line in R2
y, perpendicular to x,
and having the angle ϕ + π/2 with the horizontal axis. Parametrizing it by the
514 Sergei Kuksin
length-coordinate l we get on Σ∗ the coordinates (r, l, ϕ) ∈ R+ × R × S1, S1 =
R/2πZ, with the coordinate mapping
Φ : (r, l, ϕ) 7→
(
x = (r cosϕ, r sinϕ), y = (−l sinϕ, l cosϕ)
)
(this map is singular at r = 0). Since
|∂Φ/∂r|2 = 1, |∂Φ/∂l|2 = 1, |∂Φ/∂ϕ|2 = r2 + l2,
〈∂Φ/∂r, ∂Φ/∂l〉 = 〈∂Φ/∂r, ∂Φ/∂ϕ〉 = 〈∂Φ/∂l, ∂Φ/∂ϕ〉 = 0,
then in these coordinates the volume element on Σ∗ reeds as
√
r2 + l2 dr dl dϕ,
and the measure |z|−1dΣ∗ — as dr dl dϕ. Consider the fibre
Π : R2
x × R2
y ⊃ Σ∗ → R2
x, (x, y) 7→ x.
It has a singular fibre Π−10 = {0} × R2
y, and for any non-zero x the fibre Π−1x
equals {x} × x⊥, where x⊥ is the line in R2
y, perpendicular to x. Since dx =
r dr dϕ, then the given above presentation for the measure |z|−1dΣ∗ implies that
its restriction to the regular part Σ+
∗ of the fibred manifold Σ∗, Σ+
∗ = Σ∗ \ ({0}×
R2
y), disintegrates by the foliation Π as
(|z|−1dΣ∗) |Σ+
∗
= |x|−1 dx dx⊥y, x 6= 0, y ∈ x⊥, (2.6)
where dx⊥ is the length on the euclidean line x⊥ ⊂ R2
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|−1dΣ∗ , restricted to Σ+
∗ , disintegrates as
pd(x, y) dx dx⊥y, x ∈ Rd \ {0}, y ∈ x⊥, (2.7)
where x⊥ is the orthogonal complement to x in Rdy, , dx⊥ is the volume element
on this Euclidean space, and the function pd satisfies the estimate pd ≤ C(|x| +
|y|)d−2|x|1−d.
3. 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
Kδ = {z = (x, y) : |x| ≤ δ, |y| ≤ δ} ⊂ Rd × Rd.
An upper bound for the integral over Kδ follows from Lemma 2.1 of [2]:
| 〈|Jν |,Kδ〉 | ≤
∫
Kδ
|F (z)| dz
(x · y)2 + ν2
≤ Cν−1δ2d−2. (3.1)
Asymptotic Properties of Integrals of Quotients. . . 515
Now we estimate the integral over the neighbourhood Σnbh of Σ∗. For this
end, using (2.3), for 0 ≤ A < B ≤ ∞ we disintegrate
〈
Jν , (Σ
nbh)BA
〉
as
〈
Jν , (Σ
nbh)BA
〉
=
∫
Σ1
m(dη)
∫ B
A
dt t2d−1
∫ θ0
−θ0
dθ
F (η, t, θ)µ(η, θ) cos(λx · y)
t4θ2 + ν2
=
∫
Σ1
m(dη)
∫ B
A
dt t2d−1Υν(η, t), (3.2)
where
Υν(η, t) = t−4
∫ θ0
−θ0
F (η, t, θ)µ(η, θ) cos(λt2θ) dθ
θ2 + ε2
, ε = νt−2.
To study Υν we first consider the integral Υ0
ν , obtained from Υν by freezing
Fµ at θ = 0. Since µ(η, 0) = 1, then
Υ0
ν = 2t−4F (η, t, 0)
∫ θ0
0
cos(λt2θ) dθ
θ2 + ε2
= 2ν−1t−2F (η, t, 0)
∫ θ0/ε
0
cos(νλw) dw
w2 + 1
.
Consider the integral
2
∫ θ0/ε
0
cos(νλw) dw
w2 + 1
= 2
∫ ∞
0
cos(νλw) dw
w2 + 1
− 2
∫ ∞
θ0/ε
cos(νλw) dw
w2 + 1
=: I1 − I2.
Since
2
∫ ∞
0
cos(ξw) dw
w2 + 1
=
∫ ∞
−∞
eiξw dw
w2 + 1
= πe−|ξ|,
then I1 = πe−νλ. For I2 we have an obvious bound |I2| ≤ 2ε/θ0 = C1νt
−2. So
Υ0
ν(η, t) = πν−1t−2F (η, t, 0)(e−νλ + ∆t), |∆t| ≤ Cνt−2. (3.3)
Now we estimate the difference between Υν and Υ0
ν . Writing (Fµ)(η, t, θ) −
(Fµ)(η, t, 0) as A(η, t)θ + B(η, t, θ)θ2, where |A|, |B| ≤ C 〈t〉−N in view of (2.5),
we have
Υν −Υ0
ν = t−4
∫ θ0
−θ0
(Aθ +Bθ2) cos(λt2θ) dθ
θ2 + ε2
.
Since the first integrand is odd in θ, then its integral vanishes, and
|Υν −Υ0
ν | ≤ C 〈t〉
−N t−4
∫ θ0
−θ0
θ2 dθ
θ2 + ε2
≤ 2C 〈t〉−N t−4θ0.
So by (3.3)∣∣∣Υν(η, t)− πν−1t−2F (η, t, 0)e−νλ
∣∣∣
≤ C 〈t〉−N
(
t−4 + ν−1t−2 νt−2
)
≤ C ′ 〈t〉−N t−4. (3.4)
516 Sergei Kuksin
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∣∣∣∣〈Jν ,(Σnbh
)∞
δ
〉
− πν−1e−νλ
∫
Σ1
mdη
∫ ∞
δ
dt t2d−3F (η, t, 0)
∣∣∣∣
≤ C
∫ ∞
δ
t2d−5 〈t〉−N dt ≤ C1χd(δ).
2) Since d ≥ 2 and N > 2d− 2, then by estimate (2.5) the integral∫
Σ1
mdη
∫ ∞
0
dt t2d−3F (η, t, 0)
converges absolutely, and by (2.2) it equals∫
Σ1
mdη
∫ ∞
0
dt t2d−3F (η, t, 0) =
∫
Σ∗
|z|−1F (z) dΣ∗z.
3) Applying 1) and 2) to F replaced by F0 = C ′〈z〉−N and using that |F | ≤
|F0| by (1.1) we find that the integral
〈
Jν ,
(
Σnbh
)∞
δ
〉
also converges absolutely.
4) As |π(ξ, θ)| ≤
√
2 |ξ|, then (Σnbh)δ0 ⊂ S
√
2δ
0 ⊂ K√2δ. Therefore by (3.1)
∣∣∣∣〈Jν ,(Σnbh
)δ
0
〉
− πν−1e−νλ
∫
Σ1
mdη
∫ δ
0
dt t2d−3F (η, t, 0)
∣∣∣∣
≤
〈
|Jν |,K√2δ
〉
+ πν−1e−νλ
∫
Σ1
mdη
∫ δ
0
dt t2d−3|F (η, t, 0)|
≤ C1ν
−1δ2d−2 + C2ν
−1δ2,
for any 0 < δ ≤ 1. Choosing δ =
√
ν, from here and 1)–3) we find that∣∣∣∣〈Jν ,Σnbh
〉
− πν−1e−νλ
∫
Σ1
mdη
∫ ∞
0
dt t2d−3F (η, t, 0)
∣∣∣∣ ≤ Cχd(ν),
and that the integral
〈
Jν ,Σ
nbh
〉
converges absolutely.
5) Finally, let us estimate the integral over R2d \ Σnbh:〈
|Jν |,R2d \ Σnbh
〉
≤
∫
{|z|≤
√
ν}
|F | dz
ω2 + ν2
+ Cd
∫ ∞
√
ν
dr r2d−1
∫
Sr\Σnbh
|F (z)| dSr
ω2 + ν2
.
By item 3) of Lemma 2.1, |ω| ≥ Cr2 in Sr \ Σnbh. Jointly with (3.1) this implies
that 〈
|Jν |,R2d \ Σnbh
〉 ∣∣ ≤ C + C
∫ ∞
√
ν
r2d−1r−4 〈r〉−N dr ≤ C1χd(ν).
Asymptotic Properties of Integrals of Quotients. . . 517
So the integral Jν converges absolutely and, in view of 2) and 4),∣∣∣∣Jν − πν−1e−νλ
∫
Σ1
mdη
∫ ∞
0
dt t2d−3F (η, t, 0)
∣∣∣∣
=
∣∣∣∣Jν − πν−1e−νλ
∫
Σ∗
|z|−1F (z) dΣ∗z
∣∣∣∣ ≤ Cχd(ν). (4.1)
This proves Theorem 1.1.
5. 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 J0; everywhere else it suffices to assume that N > 2d − 4. Accordingly,
if F satisfies (1.1) with N > 2d − 4 and 〈|F |,Σ∞1 〉 < ∞, then (1.5) holds, since
〈|F |,Σ1
0〉 <∞, see Step 4) Section 4.
ii) Our approach does not apply to study integrals (1.4), where the divisor
(x · y)2 + ν2 is replaced by (x · y)2 + (νΓ(x, y))2 and Γ 6= Const. But it applies to
integrals
Jsν =
∫
R2d
dz
F (z) sin(λx · y)
(x · y)2 + ν2
,
under certain restrictions on λ. E.g., if 1 ≤ λ ≤ ν−1 and d ≥ 3, then Jsν = 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 7→ x · y is replaced by any non-degenerate indefinite quadratic form of z ∈ RM ,
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
J ′ν =
∫
R2
F (x, y) cos(λxy)
x2y2 + ν2
dxdy,
where F ∈ C2
0 (R2) 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: C1 = {(x, y) :
y = 0, x > 0}. Now the coordinate ξ is a point in R+ with (xξ, yξ) = (ξ, 0) and
with the normal N(ξ) = (0, ξ), the set Σ1 ∩ C1 is the single point (1, 0) and the
coordinate (η, t, θ) in the vicinity of C1 degenerates to (t, θ), t > 0, |θ| < θ0,
with the coordinate-map (t, θ) 7→ (t, tθ). The relations (2.2) and (2.3) are now
518 Sergei Kuksin
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
∣∣J ′ν − πν−1e−νλ
∫
Σ′
F (z)
|z|
dz
∣∣ ≤ C,
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.
References
[1] S.Yu. Dobrokhotov, V.E. Nazaikinskii, and A.V. Tsvetkova, On an approach to
the computation of the asymptotics of integrals of rapidly varying functions, Mat.
Zametki 103 (2018), 680–692 (Russian); Engl. transl.: Math. Notes 103 (2018),
713–723.
[2] S. Kuksin, Asymptotic expansions for some integrals of quotients with degenerated
divisors, Russ. J. Math. Phys. 24 (2017), 476–487.
[3] S. Nazarenko, Wave Turbulence, Lecture Notes in Physics, 825, Springer, Heidel-
berg, 2011.
Received February 1, 2018.
Sergei Kuksin,
Institut de Mathémathiques de Jussieu–Paris Rive Gauche, CNRS, Université Paris
Diderot, UMR 7586, Sorbonne Paris Cité, F-75013, Paris, France;
School of Mathematics, Shandong University, Shanda Nanlu, 27, 250100, PRC;
Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia,
E-mail: Sergei.Kuksin@imj-prg.fr
Асимптотичнi властивостi iнтегралiв вiд часток,
коли чисельник осцiлює, а знаменник вироджується
Sergei Kuksin
Ми вивчаємо асимптотичне поводження при ν → 0 iнтегралiв в R2d =
{(x, y)} вiд виразiв вигляду F (x, y) cos(λx · y)
/(
(x · y)2 + ν2
)
, де λ ≥ 0 i F
досить швидко спадає на нескiнченностi. Подiбнi iнтеграли виникають
в теорiї хвильової турбулентностi.
Ключовi слова: асимптотичнi iнтеграли, iнтеграли, що осцiлюють,
чотирихвильовi взаємодiї.
mailto:Sergei.Kuksin@imj-prg.fr
Introduction
Geometry of the quadric {xy=0} and its vicinity
The geometry of the quadric.
The volume element d * and the measure |z|-1d *.
Integral over the vicinity of
End of the proof of Theorem 1.1
Comments
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