Singular integrals associated with functions of finite type and extrapolation

Abstract We consider a singular integral along a submanifold of finite type. We prove a certain Lp estimate for the singular integral, which is useful in applying an extrapolation method that shows Lp boundedness of the singular integral under a sharp condition of the kernel.

Let a function in L 1 (S n−1 ) satisfy S n−1 (θ) dσ(θ) = 0, (1.1) where dσ denotes the Lebesgue surface measure on the unit sphere S n−1 in R n . Throughout this note we assume n ≥ 2. Let s , s ≥ 1, denote the collection of functions h on R + = {t ∈ R : t > 0} satisfying where Z denotes the set of integers. We define where the supremum is taken over all s and R such that |s| < tR/2 (see [6,12]). For η > 0, let η denote the family of functions h satisfying h η = sup t∈(0, 1] t −η ω(h, t) < ∞.
In the previous works, the operator T was studied under the condition that h is a constant function. In this note, we consider the operator T under a more general condition on h. We shall prove the following: Theorem 1.1 Let q ∈ (1,2], ∈ L q (S n−1 ) and h ∈ η 1 for some η > 0. Suppose that satisfies the condition (1.1). Let T be defined as in (1.2). Then we have for all p ∈ (1, ∞), where the constant C p is independent of q, h and .
Then, as an application of Theorem 1.1 and extrapolation, we have the following theorem. Theorem 1.2 Let h ∈ η 1 for some η > 0. Suppose that is in L log L(S n−1 ) and satisfies the condition (1.1). Let T be as in (1.2). Then we have The extrapolation argument that proves Theorem 1.2 from Theorem 1.1 can be found in [8,9,10,11] (see also [15, Chap. XII, pp. 119-120]). If the function h is assumed to be a constant function in Theorem 1.2, we have a result of Al-Salman and Pan shown in [1] (see [1, Theorem 1.1]); so we can give a different proof of the result by applying Theorem 1.1 and extrapolation. Relevant results can be found in [8,9,10,11].
In Section 2, we shall prove Theorem 1.1. Consider a singular integral of the form where P(y) is a polynomial mapping from R n to R d satisfying P(−y) = −P(y) (P = 0), h ∈ s for s ∈ (1, 2] and is a function in L q (S n−1 ), q ∈ (1, 2], satisfying (1.1). Then, it has been proved that for all p ∈ (1, ∞), where the constant C p is independent of q, s, , h and the polynomial components of P if they are of fixed degree (see [8,Theorem 1]). Outline of our proof of Theorem 1.1 is similar to that of the proof for [8,Theorem 1]. We apply methods of [4] to obtain some basic estimates. We need to assume that h ∈ η 1 for some η > 0 to prove certain Fourier transform estimates. As in [8] (see also [9,10]), a key idea of the proof of Theorem 1.1 is to apply a Littlewood-Paley decomposition adapted to an appropriate lacunary sequence depending on q for which ∈ L q (S n−1 ).
In Section 3, we shall give analogs of Theorems 1.1 and 1.2 for a maximal singular integral operator related to T . In what follows we also write f L p (R d ) = f p and L q (S n−1 ) = q . Throughout this note, the letter C will be used to denote nonnegative constants which may be different in different occurrences.
Let T be as in Theorem 1.

where B is as in Proposition 2.1 and the constant C is independent of q
We can easily derive Theorem 1.1 from Proposition 2.2. Proposition 2.1 is used to prove Proposition 2.2. To prove Proposition 2.2 we also need the following.
for all k ∈ Z satisfying k ≤ L with some constants c i (1 ≤ i ≤ 3), where L is a negative integer, L ≤ −4, which will be determined in Lemma 2.4 below.
To prove Lemma 2.3 we need the following two lemmas.
and let σ k be as in (2.1). Then, there exist a positive integer M, a positive number 0 ∈ (0, 1/4) and a negative integer L, L ≤ −4, such that The constants M, 0 , L and C are independent of ρ, q, h and .
and ∈ L q (S n−1 ). Let P be a real-valued polynomial on R n of degree m ≥ 1. Write Then there exists a constant C > 0 independent of ρ, k, q, h, and the coefficients of the polynomial P such that We can prove Lemma 2.5 similarly to the proof of Lemma 2.4 of [4]. To prove Lemma 2.4 we need the following two results, which can be found in [4].

Lemma 2.6 Let
: B(0, 1) → R d be smooth and of finite type at the origin. Define Then, there exist constants R, δ ∈ (0, 1/4) and a mapping from S d−1 to a finite set of positive integers such that Suppose that ϕ is compactly supported and that where k is a positive integer. Then, there exists a positive constant C depending only on k and ϕ such that Define a function F on an appropriate subinterval of R + by F(t) = ξ, (tx) for fixed ξ ∈ S d−1 and x ∈ B(0, 1). Then, we note that Proof of Lemma 2.4: Take an integer ν ≥ 1 and a ∈ [2,4] such that ρ = a ν . Let , δ, R and (ξ) be as in Lemma 2.6. Put 0 = max ξ∈S d−1 (ξ). Let L be a negative integer such that We take u = (|ξ|ρ kM ) −ζ/q for a suitable M with M ≥ 0 and ζ > 0, which will be specified below. We assume |ξ|ρ kM ≥ 1 for the moment. Define Then, by (2.8) where we have used the fact that ν ≈ log ρ. By Lemma 2.7

Proof of Lemma 2.5: Let
Let a ∈ [2,4] and ν ≥ 1 be as in the proof of Lemma 2.4. Decompose

Let h j (t) = s<t/2 h(ρ k a j (t − s))φ u (s) ds be as in the proof of Lemma 2.4 and
Now, we assume that b := ρ km |α|=m |a α | ≥ 1 and put u = (a jm b) −1/(4mq ) . Then, as in the proof of Lemma 2.4, an integration by parts argument implies that and hence by Hölder's inequality and [7, Corollary 1] By this estimate and (2.11) we see that if ρ km |α|=m |a α | ≥ 1. Along with (2.12), this implies the conclusion of Lemma 2.5.

Proof of Proposition 2.2: LetT
where the constants c j are independent of β m+1 . This is possible since β m+1 ≥ 2. Let We also write S j . Plancherel's theorem and the estimates (2.5)-(2.7) imply that By (2.5), (2.21) and the proof of Lemma in [3, p. 544], we have the following.

Proof: Define
are as in the proof of Proposition 2.2). Then, j . Arguing as in the proof of (2.19), and using Plancherel's theorem and the estimates (2.27)-(2.30), we have This proves the assertion of Lemma 2.9 for j = 1.
We now assume the estimate of Lemma 2.9 for j = s and prove it for j = s + 1. By induction, this will complete the proof of Lemma 2.9. From the estimate (2.31), it follows that k | * f |. By our assumption we have g m ( f ) p s ≤ C AB 2/ p s f p s . This estimate and (2.26) imply Arguing as in the proof of (2.25), and using (2.27), (2.33) and (2.34), we can now obtain the estimate of Lemma 2.9 for j = s + 1. This completes the proof of Lemma 2.9.
Let p ∈ (1 + θ, 2] and let { p j } ∞ 1 be as in Lemma 2.9. Then, we can find a positive integer N such that p N+1 < p ≤ p N . The estimate (2.32) now follows from interpolation between the estimates of Lemma 2.9 for j = N and j = N + 1. This finishes the proof of (2.4) for j = m. By induction, this completes the proof of Proposition 2.1.
Proof of Theorem 1.1: By taking ρ = 2 q in Proposition 2.2 we see that

Estimates for maximal functions
Let for all p ∈ (1, ∞), where C p is independent of q, h and .
By Theorem 3.1 and extrapolation we have the following result.

Theorem 3.2 Let
∈ L log L(S n−1 ) and h ∈ η 1 for some η > 0. Suppose that satisfies the condition (1.1). Let T * f be defined as in (3.1) with the functions h and . Then for all p ∈ (1, ∞).
If the function h is identically 1, then Theorem 3.2 was shown in [1]. To prove Theorem 3.1, we use the following result. This can be proved by results in Section 2. where k ≤ L and δ = δ 0 is the delta function on R d (see [3,5]). Then, we have Then, we have (3.8) where J ρ ( f )(x) = ρ L+1 ≤|y|<1 | f(x − (y))||K(y)| dy. We note that