Uniform boundedness of oscillatory singular integrals with rational phases

Hussain Al-Qassem; Leslie Cheng; Yibiao Pan

doi:10.1515/forum-2024-0077

Article Open Access

Uniform boundedness of oscillatory singular integrals with rational phases

Hussain Al-Qassem , Leslie Cheng and Yibiao Pan

Published/Copyright: January 13, 2025

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From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

We prove the uniform boundedness of oscillatory singular integrals with singular kernels | x | - n ⁢ Ω ⁢ ( x | x | ) and rational phases of the form P ⁢ ( x ) + 1 Q ⁢ ( x ) for arbitrary real-valued polynomials P and Q. Our main result shows that the condition Q ⁢ ( 0 ) = 0 imposed in [M. Folch-Gabayet and J. Wright, An estimation for a family of oscillatory integrals, Studia Math. 154 2003, 1, 89–97] is superfluous, which answers a question left open in that paper. As a secondary improvement of existing results, we also extend the space for Ω ⁢ ( ⋅ ) from L ⁢ log ⁡ L ⁢ ( 𝕊 n - 1 ) to the strictly larger space H 1 ⁢ ( 𝕊 n - 1 ) .

Keywords: Oscillatory integrals; singular integrals; Calderón–Zygmund kernels; Hardy spaces

MSC 2020: 42B20; 42B30; 42B35

1 Introduction

The investigation of oscillatory singular integrals has a rich and enduring history ([1, 4, 6, 7, 8, 10, 11]). For oscillatory singular integrals in dimensions higher than 1, boundedness does not hold for general rational phases. In [5], the authors obtained some very interesting estimates for oscillatory singular integrals with phase functions of the form P ⁢ ( x ) + 1 Q ⁢ ( x ) , where P ⁢ ( x ) and Q ⁢ ( x ) are real-valued polynomials in n variables. To describe their results, we let n ≥ 2 , K ⁢ ( x ) be a Calderón–Zygmund kernel given by

(1.1) K ⁢ ( x ) = Ω ⁢ ( x | x | ) | x | n ,

where Ω : 𝕊 n - 1 → ℂ is integrable over the unit sphere 𝕊 n - 1 with respect to the induced Lebesgue measure σ and satisfies

(1.2) ∫ 𝕊 n - 1 Ω ⁢ ( x ) ⁢ 𝑑 σ ⁢ ( x ) = 0 .

Let d ∈ ℕ ∪ { 0 } , and let 𝒫 n , d denote the space of polynomials in n variables whose coefficients are real and whose degrees do not exceed d. The following is a result from [5].

Theorem 1.1 (Folch-Gabayet and Wright [5]).

Let K ⁢ ( x ) be a Calderón–Zygmund kernel given by (1.1)–(1.2). Let P ⁢ ( x ) , Q ⁢ ( x ) ∈ P n , d such that Q ⁢ ( 0 ) = 0 and Ω ∈ L ⁢ log ⁡ L ⁢ ( S n - 1 ) . Then

(1.3) | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( x ) + 1 Q ⁢ ( x ) ) ⁢ K ⁢ ( x ) ⁢ 𝑑 x | ≤ B ,

where B may depend on ∥ Ω ∥ L ⁢ log ⁡ L ⁢ ( S n - 1 ) , n and d but not otherwise on the coefficients of P and Q.

This naturally led to the following question:

Question.

Would the conclusion of Theorem 1.1 hold if the condition Q ⁢ ( 0 ) = 0 is removed?

For the case deg ⁡ ( Q ) = 1 , the authors of [5] answered the above question in the affirmative:

Theorem 1.2 ([5]).

Let K ⁢ ( x ) be a Calderón–Zygmund kernel given by (1.1)–(1.2). Let P ⁢ ( x ) ∈ P n , d and Q ⁢ ( x ) = a + v ⋅ x , where a ∈ R and v ∈ R n . Suppose that Ω ∈ L ⁢ log ⁡ L ⁢ ( S n - 1 ) . Then

(1.4) | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( x ) + 1 Q ⁢ ( x ) ) ⁢ K ⁢ ( x ) ⁢ 𝑑 x | ≤ B ,

where B may depend on ∥ Ω ∥ L ⁢ log ⁡ L ⁢ ( S n - 1 ) , n and d but not otherwise on a, v and the coefficients of P.

The main purpose of this paper is to give a complete answer to the question stated above by showing that the condition Q ⁢ ( 0 ) = 0 in Theorem 1.1 can be dropped irrespective of the degree of Q ⁢ ( x ) .

In addition to the improvement of Theorem 1.1 by lifting the vanishing condition on Q ⁢ ( 0 ) , we shall also expand the class of K ⁢ ( x ) in Theorem 1.1 by allowing Ω ⁢ ( ⋅ ) to be in H 1 ⁢ ( 𝕊 n - 1 ) , the Hardy space over the unit sphere. It is well known that the space L ⁢ log ⁡ L ⁢ ( 𝕊 n - 1 ) is a proper subspace of H 1 ⁢ ( 𝕊 n - 1 ) . We state our result as follows.

Theorem 1.3.

Let K ⁢ ( x ) be a Calderón–Zygmund kernel given by (1.1)–(1.2). Let P ⁢ ( x ) , Q ⁢ ( x ) ∈ P n , d and Ω ∈ H 1 ⁢ ( S n - 1 ) . Then

(1.5) | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( x ) + 1 Q ⁢ ( x ) ) ⁢ K ⁢ ( x ) ⁢ 𝑑 x | ≤ B ⁢ ∥ Ω ∥ H 1 ⁢ ( 𝕊 n - 1 ) ,

where B may depend on n and d but not otherwise on the coefficients of P and Q.

The proof of Theorem 1.3 will appear in Section 3.

Boundedness results such as (1.5) can be used together with Plancherel’s Theorem to obtain the L 2 boundedness of corresponding singular integral operators defined by polynomial mappings. We refer the readers to [5] for more details.

In the rest of the paper we shall use A ≲ B ( A ≳ B ) to mean that A ≤ c ⁢ B ( A ≥ c ⁢ B ) for a certain constant c whose actual value is not essential for the relevant arguments to work. We shall also use A ≈ B to means “ A ≲ B and B ≲ A ”.

2 A few lemmas

In order to prove Theorem 1.3, one of the tools we shall need is the following lemma:

Lemma 2.1.

Let A > 1 , d ∈ N and

q ⁢ ( t ) = ∑ j = 1 d q j ⁢ t j ,

where q 1 , … , q d ∈ R and q d ≠ 0 . Then there are m ( m ≤ d ) disjoint subintervals G 1 = ( L 1 , R 1 ) , … , G m = ( L m , R m ) of ( 0 , ∞ ) such that

0 = L 1 < R 1 < L 2 < R 2 < ⋯ < L m < R m = ∞ ,
for each l ∈ { 1 , … , m } , there exists a k l ∈ { 1 , … , d } such that

| q k l ⁢ t k l | > A ⋅ max ⁡ { | q k ⁢ t k | : k ∈ { 1 , … , d } \ { k l } }

for all t ∈ G l ,
for every ξ ∈ { L 2 , … , L m , R 1 , … , R m - 1 } , there exists a pair of j , k ∈ { 1 , … , d } such that j ≠ k and ξ ≈ | q k q j | 1 j - k ,
for 1 ≤ l ≤ m - 1 ,

L l + 1 R l ≤ A d ⁢ ( d - 1 ) 2 .

The above lemma can be viewed as a“strengthening” of [5, Lemma 2.1]. For the proof, instead of employing the method of induction as done in [5], we shall use a more direct approach.

Proof.

Let Λ = { j : 1 ≤ j ≤ d ⁢ and ⁢ q j ≠ 0 } . Since (i)–(iv) hold trivially when | Λ | = 1 , we may assume that | Λ | ≥ 2 . For every j ∈ Λ , let

S j = { t ∈ ( 0 , ∞ ) : | q j t j | > A ⋅ max { | q k t k | : k ∈ Λ \ { j } } .

Then either S j = ∅ or S j = ( a j , b j ) where

a j = max ⁡ ( { ( A ⁢ | q k q j | ) 1 j - k : k ∈ Λ ⁢ and ⁢ k < j } ∪ { 0 } ) ,

b j = min ⁡ ( { ( A ⁢ | q k q j | ) 1 j - k : k ∈ Λ ⁢ and ⁢ k > j } ∪ { ∞ } ) ,

and a j < b j . By A > 1 ,

S j ∩ S j ′ = ∅

for any j , j ′ ∈ Λ satisfying j ≠ j ′ . Let G 1 , … , G m denote all the nonempty S j ’s, arranged from left to right and let G l = ( L l , R l ) for 1 ≤ l ≤ m . Clearly, (i)–(iii) are satisfied.

Let l ∈ { 1 , … , m - 1 } . Then there exist an integer s satisfying 1 ≤ s ≤ d ⁢ ( d - 1 ) 2 and a partition

R l = ζ 0 < ζ 1 < ⋯ < ζ s = L l + 1 ,

such that

| q j ⁢ t j | ≠ | q k ⁢ t k |

for all distinct j , k in Λ and t ∈ [ R l , L l + 1 ] \ { ζ 0 , ζ 1 , … , ζ s } . For each ν ∈ { 1 , … , s } , there are j ν , k ν ∈ Λ such that j ν ≠ k ν and

(2.1) max ⁡ { | q k ⁢ t k | : k ∈ Λ \ { j ν } } = | q k ν ⁢ t k ν | < | q j ν ⁢ t j ν |

for all t ∈ ( ζ ν - 1 , ζ ν ) . Since

( ζ ν - 1 , ζ ν ) ⊆ ( R l , L l + 1 ) ⊆ ( 0 , ∞ ) \ ⋃ j ∈ Λ S j ,

we have

(2.2) | q j ν ⁢ t j ν | ≤ A ⋅ max ⁡ { | q k ⁢ t k | : k ∈ Λ \ { j ν } } = A ⁢ | q k ν ⁢ t k ν |

for all t ∈ ( ζ ν - 1 , ζ ν ) . It follows from (2.1) and (2.2) that, for ν = 1 , … , s ,

ζ ν ζ ν - 1 ≤ A 1 | j ν - k ν | ≤ A ,

which implies (iv). ∎

Next we recall the classical van der Corput’s lemma.

Lemma 2.2.

Let ϕ be a real-valued C k function on [ a , b ] satisfying | ϕ ( k ) ⁢ ( x ) | ≥ 1 for every x ∈ [ a , b ] . Suppose that k ≥ 2 , or that k = 1 and ϕ ′ is monotone on [ a , b ] . Then there exists a positive constant c k such that

| ∫ a b e i ⁢ λ ⁢ ϕ ⁢ ( x ) ⁢ 𝑑 x | ≤ c k ⁢ | λ | - 1 k

for all λ ∈ ℝ . The constant c k is independent of λ , a , b and ϕ.
Let ϕ and c k be the same as in (i). If ψ ∈ C 1 ⁢ ( [ a , b ] ) , then

| ∫ a b e i ⁢ λ ⁢ ϕ ⁢ ( x ) ⁢ ψ ⁢ ( x ) ⁢ 𝑑 x | ≤ c k ⁢ | λ | - 1 k ⁢ ( ∥ ψ ∥ L ∞ ⁢ ( [ a , b ] ) + ∥ ψ ′ ∥ L 1 ⁢ ( [ a , b ] ) )

holds for all λ ∈ ℝ .

Below is an easy consequence of van der Corput’s lemma which will be needed in our proof of Theorem 1.3.

Lemma 2.3.

Let Φ ⁢ ( t , u ) be a real-valued C ∞ function on [ a , b ] × U , where U is an open set in R m . Suppose that d ∈ N ∪ { 0 } and for every ( t , u ) ∈ [ a , b ] × U , there exists an integer k = k ⁢ ( t , u ) > d such that

∂ k ⁡ Φ ⁢ ( t , u ) ∂ ⁡ t k ≠ 0 .

Then, for every compact subset W of U, there exist two positive constants ρ = ρ ⁢ ( d , m , a , b , Φ , W ) and C = C ⁢ ( d , m , a , b , Φ , W ) such that

| ∫ J e i ⁢ [ R ⁢ ( t ) + λ ⁢ Φ ⁢ ( t , u ) ] ⁢ ψ ⁢ ( t ) ⁢ 𝑑 t | ≤ C ⁢ | λ | - ρ ⁢ ( ∥ ψ ∥ L ∞ ⁢ ( J ) + ∥ ψ ′ ∥ L 1 ⁢ ( J ) )

holds for all subintervals J of [ a , b ] , ψ ∈ C 1 ⁢ ( J ) , λ ∈ R , u ∈ W and R ⁢ ( ⋅ ) ∈ P 1 , d .

To prove the above lemma, one first uses Lemma 2.2 (or a direct integration by parts when d = 0 and k = 1 ) locally and then finish with a compactness argument. Details are omitted.

Another result we shall need is the following lemma from Stein [9, p. 331].

Lemma 2.4.

Let q ⁢ ( x ) be a homogeneous polynomial of degree d on R n . Write

m q = ∫ 𝕊 n - 1 | q ⁢ ( ω ) | ⁢ 𝑑 σ ⁢ ( ω ) .

Then

∫ 𝕊 n - 1 | ln ⁡ ( | q ⁢ ( ω ) m q | ) | ⁢ 𝑑 σ ⁢ ( ω ) ≤ B d ,

where B d is independent of q ⁢ ( ⋅ ) .

3 Proof of Theorem 1.3

We are now ready to present the proof of Theorem 1.3. Initially we will assume that P , Q ∈ 𝒫 n , d , Ω ∈ L ∞ ⁢ ( 𝕊 n - 1 ) and satisfies the vanishing mean value condition (1.2). We will prove that there exists an B n , d > 0 independent of Ω and the coefficients of the polynomials P and Q such that

(3.1) | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( x ) + 1 Q ⁢ ( x ) ) ⁢ Ω ⁢ ( x ) | x | n ⁢ 𝑑 x | ≤ B n , d ⁢ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Since the case Q ⁢ ( 0 ) = 0 is already covered by the result of Folch-Gabayet and Wright (see Theorem 1.1), we shall assume that

(3.2) Q ⁢ ( x ) = η ⁢ ( 1 + ∑ 1 ≤ | α | ≤ d a α ⁢ x α ) ,

where η = Q ⁢ ( 0 ) ≠ 0 . For 1 ≤ k ≤ d , let

q k ⁢ ( x ) = ∑ | α | = k a α ⁢ x α .

Then, for each k ∈ { 1 , … , d } , either q k ⁢ ( ⋅ ) ≡ 0 or q k ⁢ ( ω ) ≠ 0 for a.e. ω ∈ 𝕊 n - 1 .

Let

Λ = Λ Q = { k : 1 ≤ k ≤ d ⁢ and ⁢ q k ⁢ ( ⋅ ) ≢ 0 } .

Then, for all ω ∈ 𝕊 n - 1 and t > 0 ,

(3.3) Q ⁢ ( t ⁢ ω ) = η ⁢ ( 1 + ∑ k ∈ Λ q k ⁢ ( ω ) ⁢ t k ) .

We will present our argument for the more general case of | Λ | > 1 , while omitting the discussion of the case when | Λ | = 1 . However, it can be treated in a similar but simpler manner.

Let A = 2 ⁢ d 2 . For each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 , let

G j ⁢ ( ω ) = { t ∈ ( 0 , ∞ ) : | q j ⁢ ( ω ) | ⁢ t j > A ⁢ | q k ⁢ ( ω ) | ⁢ t k ⁢ for all ⁢ k ∈ Λ \ { j } } .

Thus,

G j ⁢ ( ω ) = ( ⋂ k ∈ Λ j + ( 0 , ( A ⁢ | q k ⁢ ( ω ) | | q j ⁢ ( ω ) | ) 1 j - k ) ) ∩ ( ⋂ k ∈ Λ j - ( ( A ⁢ | q k ⁢ ( ω ) | | q j ⁢ ( ω ) | ) 1 j - k , ∞ ) ) ,

where we used

Λ j + = { k ∈ Λ : k > j } , Λ j - = { k ∈ Λ : k < j }

and the convention that

⋂ S ∈ ∅ S = ( 0 , ∞ ) .

For each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 , let

G j ( 1 ) ⁢ ( ω ) = G j ⁢ ( ω ) ∩ ( 0 , 4 - d j ⁢ | q j ⁢ ( ω ) | - 1 j ) ,

G j ( 2 ) ⁢ ( ω ) = G j ⁢ ( ω ) ∩ [ 4 - d j ⁢ | q j ⁢ ( ω ) | - 1 j , 4 d j ⁢ | q j ⁢ ( ω ) | - 1 j ] ,

G j ( 3 ) ⁢ ( ω ) = G j ⁢ ( ω ) ∩ ( 4 d j ⁢ | q j ⁢ ( ω ) | - 1 j , ∞ ) .

For t ∈ G j ( 3 ) ⁢ ( ω ) , we have

(3.4) Q ⁢ ( t ⁢ ω ) ≈ q j ⁢ ( ω ) ⁢ t j

and

(3.5) d d ⁢ t ( Q ( t ω ) ) ≈ d d ⁢ t ( q j ( ω ) t j ) ) .

It follows from the arguments in the proof of [5, Theorem 1.1] and (3.4)–(3.5) that

(3.6) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ G j ( 3 ) ⁢ ( ω ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Trivially,

(3.7) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ G j ( 2 ) ⁢ ( ω ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Therefore, in order to prove (3.1), it suffices to prove that

(3.8) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ G j ( 1 ) ⁢ ( ω ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 )

and

(3.9) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ ( 0 , ∞ ) \ ⋃ j ∈ Λ G j ⁢ ( ω ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Since (3.9) follows from Lemma 2.1 (iv) easily, we will focus our attention on the proof of (3.8). By applying Lemma 2.3 with [ a , b ] = [ 0 , 1 4 ] , u = ( u 1 , … , u d ) ,

U = { u ∈ ℝ d : 4 5 < max 1 ≤ j ≤ d ⁡ | u j | < 6 5 } , W = { u ∈ ℝ d : max 1 ≤ j ≤ d ⁡ | u j | = 1 } ,

and

(3.10) Φ ⁢ ( t , u ) = ( 1 + ∑ k = 1 d u k ⁢ t k ) - 1 ,

there exist two positive constants ρ = ρ ⁢ ( d ) and C d such that

(3.11) | ∫ J e i ⁢ [ R ⁢ ( t ) + λ ⁢ Φ ⁢ ( t , u ) ] ⁢ ψ ⁢ ( t ) ⁢ 𝑑 t | ≤ C d ⁢ | λ | - ρ ⁢ ( ∥ ψ ∥ L ∞ ⁢ ( J ) + ∥ ψ ′ ∥ L 1 ⁢ ( J ) )

holds for all subintervals J of [ 0 , 1 4 ] , ψ ∈ C 1 ⁢ ( J ) , λ ∈ ℝ , u ∈ W and R ⁢ ( ⋅ ) ∈ 𝒫 1 , d .

Let N = [ ρ - 1 ] + 1 , δ = min ⁡ { | η | 1 N , 1 } . For each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 , let

Y j ⁢ ( ω , δ ) = G j ( 1 ) ⁢ ( ω ) ∩ ( | q j ⁢ ( ω ) | - 1 j ⁢ δ 1 j , ∞ ) ,

H j ⁢ ( ω , δ ) = G j ( 1 ) ⁢ ( ω ) ∩ [ 1 6 ⁢ | q j ⁢ ( ω ) | - 1 j ⁢ δ 1 j , | q j ⁢ ( ω ) | - 1 j ⁢ δ 1 j ] ,

Z j ⁢ ( ω , δ ) = G j ( 1 ) ⁢ ( ω ) ∩ ( 0 , 1 6 ⁢ | q j ⁢ ( ω ) | - 1 j ⁢ δ 1 j ) .

We also let ν ω ∈ Λ such that

| q ν ω ⁢ ( ω ) | 1 ν ω = max ⁡ { | q k ⁢ ( ω ) | 1 k : k ∈ Λ } .

For each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 , if Y j ⁢ ( ω , δ ) ≠ ∅ and t ∈ Y j ⁢ ( ω , δ ) , then

δ 1 j < | q j ⁢ ( ω ) | 1 j ⁢ t ≤ | q ν ω ⁢ ( ω ) | 1 ν ω ⁢ t = ( | q ν ω ⁢ ( ω ) | ⁢ t ν ω ) 1 ν ω ≤ ( | q j ⁢ ( ω ) | ⁢ t j ) 1 ν ω < 4 - d ν ω ≤ 1 4 .

It is thus clear that, in this case, δ = | η | 1 N < 1 and the set

| q ν ω ⁢ ( ω ) | 1 ν ω ⁢ Y j ⁢ ( ω , δ ) = { | q ν ω ⁢ ( ω ) | 1 ν ω ⁢ t : t ∈ Y j ⁢ ( ω , δ ) }

is a subinterval of ( δ 1 j , 1 4 ) . Let

P ~ ω ⁢ ( t ) = P ⁢ ( | q ν ω ⁢ ( ω ) | - 1 ν ω ⁢ t ⁢ ω ) ,

u = ( u 1 , … , u d ) , where

u k = { q k ⁢ ( ω ) ⁢ | q ν ω ⁢ ( ω ) | - k ν ω if ⁢ k ∈ Λ , 0 if ⁢ k ∉ Λ

and let Φ ⁢ ( ⋅ , ⋅ ) be given as in (3.10). By (3.11),

≤ C d ⁢ | η | ρ ⁢ ( 1 δ 1 j + ∫ δ 1 j ∞ d ⁢ t t 2 ) = 2 ⁢ C d ⁢ | η | ρ - 1 N ⁢ j ≲ 1 ,

which implies that

(3.12) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ Y j ⁢ ( ω , δ ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Trivially we have

(3.13) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ H j ⁢ ( ω , δ ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

Next we shall prove that

(3.14) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ Z j ⁢ ( ω , δ ) e i ⁢ ( P ⁢ ( t ⁢ ω ) + 1 Q ⁢ ( t ⁢ ω ) ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) ,

which, together with (3.12)–(3.13), would give us (3.8).

For each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 , if t ∈ Z j ⁢ ( ω , δ ) , then

| ∑ k ∈ Λ q k ⁢ ( ω ) ⁢ t k | ≤ ( 1 + ( d - 1 ) ⁢ A - 1 ) ⁢ | q j ⁢ ( ω ) | ⁢ t j ≤ 6 - j ⁢ ( 1 + ( d - 1 ) ⁢ A - 1 ) ⁢ δ < δ 4 ≤ 1 4 .

Let

P N , η ⁢ ( x ) = P ⁢ ( x ) + η - 1 ⁢ ∑ s = 0 N - 1 ( - ∑ k ∈ Λ q k ⁢ ( x ) ) s .

Then P N , η ⁢ ( x ) is a polynomial whose degree does not exceed Nd and, for each j ∈ Λ and a.e. ω ∈ 𝕊 n - 1 ,

(3.15)

≤ 4 3 ⁢ d N ⁢ | η | - 1 ⁢ max ⁡ { ( | q j ⁢ ( ω ) | ⁢ t j ) N : t ∈ Z j ⁢ ( ω , δ ) }

≤ 4 3 ⁢ d N ⁢ | η | - 1 ⁢ δ N ≲ 1 .

Recall that Stein established in [9] that, for K ⁢ ( ⋅ ) given as in (1.1) with an Ω ∈ L ∞ ⁢ ( 𝕊 n - 1 ) and { γ α : | α | ≤ M } ⊂ ℝ ,

| ∫ ε 1 ≤ | x | ≤ ε 2 e i ⁢ ( ∑ | α | ≤ M γ α ⁢ x α ) ⁢ K ⁢ ( x ) ⁢ 𝑑 x | ≤ A M ,

where A M is independent of ε 1 , ε 2 and γ α (see [9, pp. 334–335]). It follows that, for all h > 0 ,

(3.16) | p.v. ⁢ ∫ | x | ≤ h e i ⁢ P N , η ⁢ ( x ) ⁢ K ⁢ ( x ) ⁢ 𝑑 x | ≤ B d ⁢ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 )

holds with a B d independent of h and the coefficients of P N , η . Observe that each Z j ⁢ ( ω , δ ) is the intersection of no more than ( d + 1 ) intervals in the form of either ( 0 , ( | q ( ω ) / q ~ ( ω ) | γ ) or ( ( | q ( ω ) / q ~ ( ω ) | γ , ∞ ) , where q ⁢ ( ⋅ ) , q ~ ⁢ ( ⋅ ) are homogeneous polynomials of degrees not exceeding d, and | γ | ≤ 1 . Thus, one may use (3.16) and Lemma 2.4 to get

(3.17) | ∫ 𝕊 n - 1 Ω ⁢ ( ω ) ⁢ ( ∑ j ∈ Λ ∫ Z j ⁢ ( ω , δ ) e i ⁢ P N , η ⁢ ( t ⁢ ω ) ⁢ d ⁢ t t ) ⁢ 𝑑 σ ⁢ ( ω ) | ≲ ∥ Ω ∥ L ∞ ⁢ ( 𝕊 n - 1 ) .

By combining (3.17) and (3.15), we obtain (3.14). Then (3.8) follows, as does (3.1).

We will end our proof of Theorem 1.3 by providing a brief explanation of how one can derive (1.5) from (3.1).

Let β : 𝕊 n - 1 → ℂ be an arbitrary regular H 1 atom on 𝕊 n - 1 , i.e. β ⁢ ( ⋅ ) is supported in 𝕊 n - 1 ∩ B ⁢ ( p , h ) for some p ∈ 𝕊 n - 1 and h > 0 , has mean-value zero over 𝕊 n - 1 , and satisfies ∥ β ∥ ∞ ≤ h 1 - n .

Suppose that h < 1 4 . Let M be an orthogonal matrix which has p as its first row vector. Define the linear transformation L on ℝ n by

L ⁢ ( x 1 , x 2 , … , x n ) = ( x 1 , h ⁢ x 2 , … , h ⁢ x n ) ⁢ M ,

and let

Ω ~ ⁢ ( x ) = det ⁡ ( L ) ⁢ | x | n ⁢ β ⁢ ( L ⁢ x | L ⁢ x | ) | L ⁢ x | n

for x ∈ ℝ n \ { 0 } . Then Ω ~ is homogeneous of degree 0, has mean-value zero over 𝕊 n - 1 and satisfies ∥ Ω ~ ∥ ∞ ≲ 1 . By the uniform nature of (3.1) which allows it to be applied with P ⁢ ( x ) and Q ⁢ ( x ) replaced by P ⁢ ( L ⁢ x ) and Q ⁢ ( L ⁢ x ) , we get

(3.18) | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( x ) + 1 Q ⁢ ( x ) ) ⁢ β ⁢ ( x | x | ) | x | n ⁢ 𝑑 x | = | p.v. ⁢ ∫ ℝ n e i ⁢ ( P ⁢ ( L ⁢ x ) + 1 Q ⁢ ( L ⁢ x ) ) ⁢ Ω ~ ⁢ ( x ) | x | n ⁢ 𝑑 x | ≤ B n , d ⁢ ∥ Ω ~ ∥ L ∞ ⁢ ( 𝕊 n - 1 ) ≲ 1

for all h ∈ ( 0 , 1 4 ) . The above inequality also holds when h ≥ 1 4 as it follows directly from (3.1) and ∥ β ∥ ∞ ≤ h 1 - n ≲ 1 . Finally, by the atomic decomposition of H 1 ⁢ ( 𝕊 n - 1 ) (see [2, 3]) and (3.18), one obtains (1.5). The proof of Theorem 1.3 is now complete.

Communicated by Christopher D. Sogge

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Received: 2024-02-10

Revised: 2024-11-04

Published Online: 2025-01-13

Published in Print: 2025-06-01

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Keywords for this article

Oscillatory integrals; singular integrals; Calderón–Zygmund kernels; Hardy spaces

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