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Comptes Rendus
Mathématique
Chenyan Wang and Huoxiong Wu
A note on singular oscillatory integrals with certain rational phases
Volume 361 (2023), p. 363-370
https://doi.org/10.5802/crmath.418
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 363-370
https://doi.org/10.5802/crmath.418
Harmonic analysis / Analyse harmonique
A note on singular oscillatory integrals with
certain rational phases
ChenyanWanga and HuoxiongWu∗, a
a School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
E-mails: chenyanwangxmu@163.com (C. Wang), huoxwu@xmu.edu.cn (H. Wu)
Abstract. LetΩ be homogeneous of degree zero with mean value zero, P and Q real polynomials on Rn with
Q(0) = 0 and Ω ∈ B0,0q (Sn−1) for some q > 1. This note extends and improves a classical result of Stein and
Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general form∣∣∣∣p. v.∫
Rn
ei (P (x)+1/Q(x))Ω(x/|x|)|x|n d x
∣∣∣∣≤ B ,
where B depend only on ‖Ω‖
B0,0q (S
n−1), n and the degrees of P and Q, but not on their coefficients.
2020 Mathematics Subject Classification. 42B15, 42B20, 42A50, 42A45.
Funding. Supported by the NNSF of China (Nos. 12171399, 11871101).
Manuscript received 23 February 2022, accepted 1 September 2022.
1. Introduction
Let P be a polynomial on Rn of degree at most d with real coefficients and K be a homogeneous
function of degree −n on Rn , that is,
K (x) = Ω(x/|x|)|x|n ,
whereΩ is an integrable function on the unit sphere Sn−1 and satisfies
∫
Sn−1Ωdσ= 0.
In [8] Stein showed that ifΩ ∈ L∞(Sn−1), then∣∣∣p. v.∫
Rn
e i P (x)K (x)d x
∣∣∣≤Cd ,n,K , (1)
where Cd ,n,K is independent of the coefficients of P . The corresponding one-dimensional esti-
mation was obtained by Stein and Wainger in [9], see also [7] for the sharp bound. Subsequently,
Papadimitrakis and Parissis [6], Al-Qassem et al. [1] successively extended the estimate (1) to the
cases of thatΩ ∈ L logL(Sn−1) and H 1(Sn−1), the Hardy space defined on Sn−1. It is natural to ask
the following question.
Question. Can one extend the estimate (1) to phases which are general rational functions?
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
364 Chenyan Wang and Huoxiong Wu
In 2003, Folch-Gabayet and Wright [3] showed that for general rational phases, the estimate (1)
is not true. Meanwhile, they considered the rational phases of the form P (x)+1/Q(x), where P and
Q are real polynomials with Q(0) = 0, and forΩ ∈ L logL(Sn−1), obtained the following estimate:∣∣∣∣p. v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ A, (2)
where A depends on ‖Ω‖L logL(Sn−1), n and the degrees of P and Q, but not otherwise on the
coefficients of P and Q. It is well known that
L∞
(
Sn−1
)
$
⋃
r >1
Lr
(
Sn−1
)
$ L logL
(
Sn−1
)
.
Therefore, the estimate (2) essentially improved and generalized the corresponding results in
[6, 8, 9].
On the other hand, to study the mapping properties of singular integrals with rough kernels
on Lp (Rn), Jiang and Lu introduced the following block spaces B 0,νq (S
n−1) for ν > −1 and q > 1
(see [5] for the details of block spaces).
Definition 1 ([5]). A q-block on Sn−1 is an Lq -function b (1 < q ≤∞) that satisfies
supp(b) ⊆Q, (i)
‖b‖Lq (Sn−1) ≤ |Q|1/q−1, (ii)
where Q = Sn−1 ∩ {y ∈Rn : |y −ζ| < ρ for some ζ ∈ Sn−1 and ρ ∈ (0,1]}.
Definition 2 ([5]). For ν>−1 and q > 1, the block spaces B 0,νq on Sn−1 are defined by
B 0,νq
(
Sn−1
)= {Ω ∈ L1 (Sn−1) :Ω(y ′) =∑
s
λs bs (y
′), M 0,νq ({λs }) <∞
}
.
where each λs is a complex number, each bs is a q-block supported in Qs ,
M 0,νq ({λs }) =
∑
s
|λs |
{
1+
(
log+
1
|Qs |
)1+ν}
,
and
‖Ω‖B 0,νq (Sn−1) = inf
{
M 0,νq ({λs }) : Ω(x
′) =∑
s
λs bs (x
′)
}
.
It is easy to check that
B 0,ν1q
(
Sn−1
)
$B 0,ν2q
(
Sn−1
)
, ∀ ν1 > ν2 >−1.
Moreover, it follows from [4, 10] that for any q > 1,⋃
r >1
Lr
(
Sn−1
)
$B 0,νq
(
Sn−1
)⊂ H 1 (Sn−1)+L (log+ L)1+ν (Sn−1) , ∀ ν>−1,
and B 0,νq (S
n−1)* L log+ L(Sn−1) for any ν ∈ (−1,0), in particular,⋃
r >1
Lr
(
Sn−1
)
$B 0,0q
(
Sn−1
)
, L log+ L
(
Sn−1
)
$ H 1
(
Sn−1
)
,
but the relationship between B 0,0q (S
n−1) and L log+ L(Sn−1) remains open. Therefore, it is interest-
ing to establish the estimate (2) under the assumption of thatΩ ∈ B 0,0q (Sn−1), and more generally
Ω ∈ H 1(Sn−1).
In this paper, we will establish the estimate (2) provided that Ω ∈ B 0,0q (Sn−1). As for the case
of that Ω ∈ H 1(Sn−1), which is more interesting, it is still open. Our results can be formulated as
follows.
Chenyan Wang and Huoxiong Wu 365
Theorem 3. Suppose that K (x) =Ω(x)/|x|n , where Ω ∈ B 0,0q (Sn−1) is homogeneous of degree zero
with mean value zero and q > 1, P and Q are real polynomials with Q(0) = 0. Then∣∣∣∣p. v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ B , (3)
where B depends on ‖Ω‖B 0,0q (Sn−1), n and the degrees of P and Q, but not otherwise on the coefficients
of P and Q.
Remark 4. Employing the arguments in [3, Proposition 1.4], the following result shows that the
requirement Q(0) = 0 in Theorem 3 is most likely not necessary.
Theorem 5. With K and P as in Theorem 3, but now Q(x) = a + v · x, where a ∈R and v ∈Rn∣∣∣∣p.v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ B , (4)
where B depends on ‖Ω‖B 0,0q (Sn−1), n and the degree of P but not otherwise on the coefficients of P.
As an immediate consequence of Theorems 3 and 5, we have the following result.
Corollary 6. With K and Q as in Theorem 3 or 5, but now P : Rn → Rm being a polynomial
mapping, convolution with the distribution
L(φ) = p.v.
∫
Rn
φ(P (x))e i /Q(x)K (x)d x, φ ∈S (Rm),
is bounded on L2(Rm).
Employing the arguments in proving [3, Theorem 1.1], the main ingredient of the proof of
Theorem 3 or 5 in the current paper is to establish the following integral estimate, which has an
independent interest.
Proposition 7. Let Ω ∈ B 0,0q (Sn−1) for some q > 1, P (x) =
∑
|α|=d cαxα be a homogeneous polyno-
mial of degree d on Rn . Write mP =∑|α|=d |cα|. Then∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cd‖Ω‖B 0,0q (Sn−1). (5)
The rest of this paper is organized as follows. In Section 2 we will give some auxiliary lemmas.
The proofs of our main results will be given in Section 3. We remark that some ideas in our
arguments are taken from [3].
Finally, we make some conventions on notation. Throughout this paper, we denote by C a
positive constant which is independent of the main parameters, but it may vary from line to
line. Let A, B be complex-valued quantities. We use A . B or A = O(B) to denote the estimate
|A| ≤ C |B |. We use A ∼ B to denote the estimate A . B . A. For 1 ≤ γ ≤ ∞, γ′ is the conjugate
index of γ, and 1/γ+1/γ′ = 1.
2. Preliminaries
In this section, we recall and establish some auxiliary lemmas, which will be used in our argu-
ments later.
Lemma 8 ([3]). For any polynomial Q(r ) =∑dj=1 bk r k on R+, there is a finite collection {G j }Mj=1 of
disjoint intervals, called “gaps”, of R+ with M =O(1) such that
(1) The complement R+\
⋃M
j=1{G j } is the union of M −1 disjoint “dyadic” intervals, that is, the
ratio of the endpoints of such intervals is ∼ 1.
(2) For each G j , there is a k = k j ,1 ≤ k j ≤ d, such that for r ∈G j ,
|Q(r )| ∼
∣∣∣bk j ∣∣∣r k j and |Q ′ (r )| ∼ ∣∣∣bk j ∣∣∣r k j −1
366 Chenyan Wang and Huoxiong Wu
Also if L j and R j denote the left and right endpoints of G j respectively, then
(i) if R j <∞, then R j =Cd [|bl |/|bm |]1/m−l for some 1 ≤ l < m ≤ d and
(ii) if L j > 0, then L j =Cd [|br |/|bs |]1/s−r for some 1 ≤ r < s ≤ d.
Lemma 9 ([8]). Suppose φ is real-valued and smooth on (a,b), and that |φ(k)| ≥ λ > 0 for all
t ∈ (a,b). Then ∣∣∣∣∫ b
a
e iφ(t )d t
∣∣∣∣≤Ckλ−1/k
when either k ≥ 2, or k = 1 and φ′ (t ) is monotonic.
Lemma 10. Let γ > 1,Ω ∈ Lγ(Sn−1) and P (x) = ∑|α|=d cαxα be a homogeneous polynomial of
degree d on Rn . Write mP =∑|α|=d |cα|. Then∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cdγ′‖Ω‖Lγ(Sn−1). (6)
Proof. We may assume mP = 1. By the Hölder inequality,∫
Sn−1
|Ω(ω)| · ∣∣log(|P (ω)|)∣∣dσ(ω) ≤ ‖Ω‖Lγ(Sn−1) (∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω))1/γ′ .
And(∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω))1/γ′ . (∫ 1
1/2
r n−1
∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω)dr )1/γ′
.
( ∑
k>0
(
kγ
′)γ′ ∫ 1
1/2
r n−1dr
∫{
ω∈Sn−1 :r d |P (ω)|62−kγ′
} dσ(ω)
)1/γ′
6 γ
′
( ∑
k>0
kγ
′ ∫{
1
2 6 |x|61:2−kγ
′ −16 |P (x)|62−kγ′
} d x
)1/γ′
6 γ
′
( ∑
k>0
kγ
′ ∣∣∣∣{12 6 |x|6 1 : |P (x)|6 2−kγ′
}∣∣∣∣
)1/γ′
,
which reduces matters to obtaining uniform sublevel set estimates for P under the normalisation
mP = 1. Using the fact that all norms are equivalent on the space of polynomials of degree at most
d , we can find a derivative ∂α, 0 ≤ |α| ≤ d , such that 1. |∂αP (x)| on |x|6 1. Ifα= 0, then the above
sublevel sets are empty for large k and so we may assumeα> 0. In this case, using the mean value
theorem, one can show that |{|x|6 1 : |P (x)|6 2−kγ′ }|. 2−kγ′/|α| (see e.g. [2]). Thus( ∑
k>1
kγ
′ ∣∣∣∣{12 6 |x|6 1 : |P (x)|6 2−kγ′
}∣∣∣∣
)1/γ′
6
( ∑
k>1
kγ
′
2−kγ
′
/|α|
)1/γ′
≤ ∑
k>1
k2−k/d .Cd .
This implies the desired conclusion and completes the proof of Lemma 10. 
3. Proofs of Main Results
In this section, we present the proofs of Proposition 7 and Theorem 3.
Chenyan Wang and Huoxiong Wu 367
Proof of Proposition 7. SinceΩ ∈ B 0,0q (Sn−1), we know by Definition 2 that there is a decomposi-
tion:Ω(x ′) =∑s λs bs (x ′), where each bs is a q-block, supported in Qs and∑
s
|λs |
(
1+ log+ 1|Qs |
)
<∞.
Therefore,∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω) ≤∑
s
|λs |
∫
Sn−1
|bs (ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω)
≤
 ∑
|Qs |≥e−q′
+ ∑
|Qs |<e −q′
 |λs |∫
Sn−1
|bs (ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω)
=: I + I I .
Recall that for each bs , supp(bs ) ⊂ Qs and ‖bs‖Lq (Sn−1) ≤ |Qs |1/q−1 = |Qs |−1/q
′
. If |Qs | ≥ e−q ′ , we
take γ= q and obtain
γ′ ‖bs‖Lγ(Sn−1) ≤ q ′ ‖bs‖Lq (Sn−1) ≤ q ′ |Qs |−1/q
′
. 1;
if |Qs | < e−q ′ , we take γ= log |Qs |/(1+ log |Qs |), then 1 < γ< q , γ′ = log(1/|Qs |) and
γ′ ‖bs‖Lγ(Sn−1) ≤ log
1
|Qs |
‖bs‖Lq (Sn−1) |Qs |1/γ−1/q ≤ log
1
|Qs |
|Qs |−1/γ
′
. log
1
|Qs |
.
These, combining with Lemma 10, lead to
I ≤Cd
∑
|Qs |≥e−q′
|λs |q ′ ‖bs‖Lq (Sn−1).Cd
∑
|Qs |≥e−q′
|λs | ,
and for γ= log |Qs |/(1+ log |Qs |),
I I ≤Cd
∑
|Qs |<e−q′
|λs |γ′ ‖bs‖Lγ(Sn−1).Cd
∑
|Qs |>e−q′
|λs | log 1|Qs |
.
Consequently, ∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cd ∑
s
|λs |
(
1+ log 1|Qs |
)
,
which completes the proof of Proposition 7. 
Proof of Theorem 3. The arguments are completely similar to those in proving [3, Theorem 1.1].
The only difference is replacing [3, Lemma 2.2] by Proposition 7 in the current setting. For
completeness, we present the details as follows.
We may assume P (0) = 0. Using polar coordinates write the integral in (3) as
I =
∫
Sn−1
Ω(ω)
∫ ∞
0
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω),
where Q(x) = Qω(r ) = ∑d ′j=1 q j (ω)r j , P (x) = Pω(r ) = ∑dk=1 p j (ω)r k and p j , qk are homogeneous
polynomials of degree j and k. Using Lemma 8, we may write I =∑ I j ,k +O(1), where
I j ,k =
∫
Sn−1
Ω(ω)
∫
G j ∩Fk
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω).
Here {G j } and {Fk } are the “gaps” of Qω(r ) and Pω(r ), respectively. Note that although the inner
integral of I j ,k depends on ω in a complicated way, we know the form of the endpoints of G j and
Fk as given by Lemma 8 and so it is at least measurable as a function of ω. It suffices to bound
each I j ,k separately.
We have |Qω(r )| ∼ |q jl (ω)|r jl and |Q ′ω(r )| ∼ |q jl (ω)|r jl−1 on G j , for some 1 ≤ jl ≤ d ′, and
|Pω(r )|∼ |pkm (ω)|r km and |P ′ω(r )|∼ |pkm (ω)|r km−1 on Fk , for some 1 ≤ km ≤ d .
368 Chenyan Wang and Huoxiong Wu
Therefore away from where r jl+km ∼ (|pkm (ω)| · |q jl (ω)|)−1 the size of the phase φω(r ) =
Pω(r ) + 1/Qω(r ) and its derivative is understood. In fact, on R j ,k = G j ∩ Fk ∩ [C (|pkm (ω)| ·
|q jl (ω)|)−1/(km+ jl ),∞) (for C large enough), we have∣∣φω(r )∣∣∼ ∣∣pkm (ω)∣∣r km and ∣∣φ′ω(r )∣∣∼ ∣∣pkm (ω)∣∣r km−1.
An application of van der Corput’s Lemma 9 shows∣∣∣∣∣
∫
{
r ∈R j ,k :r≥Θ
} e iφω(r ) 1r dr
∣∣∣∣∣=O(1),
whereΘ= |pkm (ω)|1/km . Since we are applying Lemma 9 with k = 1, we need to first split the inte-
gration of the above integral into O(1) intervals, whereφ′ω(r ) is monotone. In the complementary
interval, r <Θ, due to the size of φω(r ) on R j ,k , we see that∣∣∣∣∣
∫
{
r ∈R j ,k :r <Θ
}
[
e iφω(r )
1
r
−1
]
dr
∣∣∣∣∣=O(1).
Therefore for the part of I j ,k over R j ,k ,∫
Sn−1
Ω(ω)
∫
R j ,k
e iφω(r )
1
r
dr dσ(ω) =
∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r<Θ
} drr dσ(ω)+O(1).
Similarly for L j ,k =G j ∩Fk ∩(−∞, δ(|pkm (ω)| · |q jl (ω)|)−1/(km+ jl )] (for δ small enough), we have∫
Sn−1
Ω(ω)
∫
L j ,k
e iφω(r )
1
r
dr dσ(ω) =
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)+O(1),
whereΛ= |q jl (ω)|1/ jl . Therefore
I j ,k =
∫
Sn−1
Ω(ω)
∫
G j ∩Fk
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω)
=
∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r ≤Θ
} drr dσ(ω)
+
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)+O(1)
(7)
and these two last integrals can be shown to be O(1) by repeatedly applying Proposition 7.
In fact, by the structures of R j ,k and L j k , the integrals in (7):∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r ≤Θ
} drr dσ(ω), and
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)
can be written in the form ∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω),
where E(ω) is the intersection of O(1) intervals of the form [a(ω), ∞) or (−∞, a(ω)] with
a(ω) ∈
{(∣∣pk1 (ω)∣∣/ ∣∣pk2 (ω)∣∣)1/(k2−k1) ,(∣∣q j1 (ω)∣∣/ ∣∣q j2 (ω)∣∣)1/( j2− j1) ,∣∣pk (ω)∣∣−1/k , ∣∣q j (ω)∣∣−1/ j ,(∣∣pk (ω)∣∣ · ∣∣q j (ω)∣∣)−1/( j+k)} .
Without loss of the generality, we may say that E(ω) is the intersection of M half infinite inter-
vals such as [a(ω), ∞) or (−∞, a(ω)]. Let us write E(ω) = [a(ω), ∞)∩E ′(ω), where a(ω), say, is
(|pk1 (ω)|/|pk2 (ω)|)1/(k2−k1) and E ′(ω) is the intersection of M −1 half infinite intervals as in E(ω).
We can then use Proposition 7 to write∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
Chenyan Wang and Huoxiong Wu 369
where A1 = (mpk1 /mpk2 )1/(k2−k1), which is independent of ω. Indeed,∣∣∣∣∫
Sn−1
Ω(ω)
∫
[a(ω),∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣
+
∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣ ,
and for the second integral, by Lemma 9 and Proposition 7, we have∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣
≤
∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]
dr
r
dσ(ω)
∣∣∣∣
≤ 1|k2 −k1|
[∫
Sn−1
|Ω(ω)|
∣∣∣∣∣log
( ∣∣pk1 (ω)∣∣
mpk1
)∣∣∣∣∣dσ(ω)+
∫
Sn−1
|Ω(ω)|
∣∣∣∣∣log
( ∣∣pk2 (ω)∣∣
mpk2
)∣∣∣∣∣dσ(ω)
]
.O(1).
For the other forms of a(ω), the argument is similar.
Similarly, we have∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩ [A2,∞)∩E ′′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
or ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩ (−∞,A2]∩E ′′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
where A2 is independent of ω, and E ′′(ω) is the intersection of M −2 half infinite intervals as in
E(ω). Continuing this process, after M iterations, we obtain that∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω)
∣∣∣∣+O(1),
where E is the intersection of M intervals of the form [A,∞) or (−∞, A], where A is independent
of ω. Therefore, by the mean value zero ofΩ on Sn−1, we have∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω) = 0.
This implies that∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω)
∣∣∣∣+O(1) =O(1),
and completes the proof of Theorem 3. 
Proof of Theorem 5. By replacing [3, Lemma 2.2] by Proposition 7, the proof is similar to the
proof of [3, Proposition 1.4]. We omit the details here. 
References
[1] H. M. Al-Qassem, L. C. Cheng, A. Fukui, Y. Pan, “Weighted Estimates for Oscillatory Singular Integrals”, J. Funct.
Spaces Appl. 2013 (2013), article no. 543748 (3 pages).
[2] A. Carbery, M. Christ, J. Wright, “Multidimensional van der Corput and sublevel set estimates”, J. Am. Math. Soc. 12
(1999), no. 4, p. 981-1015.
[3] M. Folch-Gabayet, J. Wright, “An estimation for a family of oscillatory integrals”, Stud. Math. 154 (2003), no. 1, p. 89-
97.
[4] M. Keitoku, E. Sato, “Block spaces on the unit sphere in Rn ”, Proc. Am. Math. Soc. 119 (1993), no. 2, p. 453-455.
[5] S. Lu, M. H. Taibleson, G. Weiss, Spaces Generated by Blocks, Publishing House of Beijing Normal University, 1989.
[6] M. Papadimitrakis, I. R. Parissis, “Singular oscillatory integrals on Rn ”, Math. Z. 266 (2010), no. 1, p. 169-179.
370 Chenyan Wang and Huoxiong Wu
[7] I. R. Parissis, “A sharp bound for the Stein–Wainger oscillatory integral”, Proc. Am. Math. Soc. 136 (2008), no. 3, p. 963-
972.
[8] E. M. Stein, “Oscillation integrals in Fourier analysis”, in Beijing Lectures in Harmonic Analysis (Beijing, 1984), Annals
of Mathematics Studies, vol. 112, Princeton University Press, 1986, p. 307-355.
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