L^p estimates for maximal functions along surfaces of revolution on product spaces

1 Introduction and main results

Let n, m ≥ 2, and let R^N (N = n or m) be the N-dimensional Euclidean space. Let S^N−1 be the unit sphere in R^N equipped with the normalized Lebesgue surface measure dσ = dσ(⋅). Also, let x^′ = x|x| for x ∈ Rⁿ ∖ {0}, y^′ = y/|y| for y ∈ R^m ∖ {0}.

Let K_Ω,h(x, y) = Ω(x^′, y^′) |x|^–n|y|^–m h(|x|, |y|), where h is a measurable function on R⁺ × R⁺ and Ω is an integrable function on Sⁿ⁻¹ × S^m−1 that satisfies

For suitable mappings ϕ, ψ : R⁺ → R, consider the singular integral operator TΩ,h,ϕ,ψP1,P2 defined, initially for C0∞ functions on Rⁿ⁺¹ × R^m+1, by

where (x, y) = (x, x_n+1, y, y_m+1) ∈ Rⁿ⁺¹ × R^m+1 and P₁ : Rⁿ → R, P₂ : R^m → R are two real-valued polynomials.

When P₁(u) = 0 and P₂(v) = 0, we denote TΩ,h,ϕ,ψP1,P2 by T_Ω,h,ϕ,ψ. Also, when ϕ(t) = ψ(t) = t, then T_Ω,h,ϕ,ψ (denoted by T_Ω,h) is just the classical singular integral operator introduced by Fefferman in [1] in which he obtained the L^p boundedness of T_Ω,h for all 1 < p < ∞ whenever Ω satisfies some regularity conditions and h ≡ 1. As a matter of fact, the systematic study of such operator began by Fefferman in [1], and then it was elaborated very much by Fefferman and Stein in [2]. Subsequently, the investigation of the L^p boundedness of T_Ω,h under very various conditions on Ω and h has attracted the attention of many authors. For example, it was proved in [3] that T_Ω,h is bounded on L^p(Rⁿ × R^m) for 1 < p < ∞ when Ω ∈ L(log L)²(Sⁿ⁻¹ × S^m−1) and h satisfies certain integrability-size condition. Furthermore, the authors of [3] established the optimality of the condition in the sense that the space L(log L)²(Sⁿ⁻¹ × S^m−1) cannot be replaced by L(log L)^2–ε(Sⁿ⁻¹ × S^m−1) for any 0 < ε < 2. For more information about the importance and the recent advances on the study of such operators, the readers are refereed (for instance to [1, 2, 3, 4, 5], and the references therein).

On the other side, the study of the singular integrals on product spaces along surfaces of revolution has been started. For example, if ϕ and ψ are in C²([0, ∞)), convex and increasing functions with ϕ(0) = ψ(0) = 0, then Al-Salman in [4] showed that T_Ω,1,ϕ,ψ is bounded on L^p(Rⁿ⁺¹ × R^m+1) (1 < p < ∞) provided that Ω ∈ L(log L)²(Sⁿ⁻¹ × S^m−1). Recently, Al-Salman improved this result in [6]. In fact, when ϕ, ψ are given as in [4], he verified the L^p boundedness of T_Ω,h,ϕ,ψ for all p ∈ (1, ∞) under the conditions Ω ∈ L(log L)(Sⁿ⁻¹ × S^m−1) and h∈L2(R+×R+,drdtrt) with hL2(R+×R+,drdtrt)≤1.

The maximal operator that related to our singular integral operator is MΩ,ϕ,ψP1,P2 that given by

where U=h∈L2(R+×R+,drdtrt);hL2(R+×R+,drdtrt)≤1.

Again, when P₁(u) = 0 and P₂(v) = 0, we denote MΩ,ϕ,ψP1,P2 by 𝓜_Ω,ϕ,ψ. Also, when ϕ(t) = ψ(t) = t, then 𝓜_Ω,ϕ,ψ reduces to the classical maximal operator denoted by 𝓜_Ω. Historically, The operator 𝓜_Ω was introduced by Ding in [7] in which he proved the L² boundedness of 𝓜_Ω whenever Ω ∈ L(log L)²(Sⁿ⁻¹ × S^m−1). This result was improved independently by Al-Qassem and Pan in [8] and by Al-Salman in [9]. Precisly, they showed that 𝓜_Ω is of type (p, p) for all p ≥ 2 provided that Ω ∈ L(log L)(Sⁿ⁻¹ × S^m−1). Moreover, they pointed out that the condition Ω ∈ L(log L)(Sⁿ⁻¹ × S^m−1) is optimal in the sense that the exponent 1 in L(log L)(Sⁿ⁻¹ × S^m−1) cannot be replaced by any smaller positive number τ < 1 so that 𝓜_Ω is bounded on L²(Rⁿ⁺¹ × R^m+1). Also, an improvement of the result in [7] was obtained by Al-Qassem in [10]. Indeed, Al-Qassem established the L^p(2 ≤ p < ∞) estimates for the class 𝓜_Ω whenever Ω belongs to the block space Bq(0,0) (𝓢ⁿ⁻¹ × 𝓢^m−1) for some q > 1. Furthermore, he proved that the condition Ω ∈ Bq(0,0) (𝓢ⁿ⁻¹ × 𝓢^m−1) is nearly optimal in the sense that the operator 𝓜_Ω may lose the L² boundedness if Ω is assumed to be in the space Bq(0,ε) (𝓢ⁿ⁻¹ × 𝓢^m−1) for some –1 < ε < 0. Recently, it was found in [6] that the maximal operator 𝓜_Ω,ϕ,ψ is bounded on L^p(Rⁿ⁺¹ × R^m+1) for any p ≥ 2 if Ω ∈ L(log L)(Sⁿ⁻¹ × S^m−1), and ϕ, ψ are in C²([0, ∞)), convex and increasing functions with ϕ(0) = ψ(0) = 0. Very recently, when ϕ(t) = ψ(t) = t, Al-Dolat and et al. found in [11] that the L^p (p ≥ 2) boundedness of MΩ,ϕ,ψP1,P2 is obtained under the condition Ω ∈ L(log L)(𝓢ⁿ⁻¹ × S^m−1) ∪ Bq(0,0) (𝓢ⁿ⁻¹ × S^m−1) with q > 1. Subsequently, the investigation of the L^p boundedness of MΩ,ϕ,ψP1,P2 under weak conditions has received much attentions from many mathematicians. For the significance of considering the integral operators MΩ,ϕ,ψP1,P2 , we refear the readers to consult [8] and [11, 12, 13], among others.

The main result of this work is formulated as follows:

We remark that by the result in Theorem 1.1 and using an extrapolation argument, we get that MΩ,ϕ,ψP1,P2 is bounded on L^p(Rⁿ⁺¹ × R^m+1) for 2 ≤ p < ∞ if Ω ∈ L(log L)(𝓢ⁿ⁻¹ × S^m−1) ∪ Bq(0,0) (𝓢ⁿ⁻¹ × S^m−1) for some q > 1.

Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.

2 Preliminary lemmas

In this section, we present and prove some lemmas used in the sequel. The first lemma can be derived by applying the same technique that Al-Qassam and Pan used in [14, pp. 64-65].

It is easy to prove the above lemma by using Lemma 2.1 and the inequality 𝓝_Ω,ϕ,ψ f(x, y) ≤ 𝓝_Ω,ψ ∘ 𝓝_Ω,ϕ f(x, y), where 𝓝_Ω,ϕ f(x, y) = 𝓝_Ω,ϕ f(⋅, y)(x), 𝓝_{Ω, ψ} f(x, y) = 𝓝_Ω,ψ f(x, ⋅)(y), and ∘ denotes the composition of operators.

A significant step toward proving Theorem 1.1 is to estimate the following Fourier transform:

We shall need the following Lemma which can be acquired by using the arguments employed in the proof of [6, Theorem 4.1] as well as [15, Theorem 1.6].

3 Proof of Theorem 1.1

The proof of Theorem 1.1 mainly depends on the approaches employed in the proof of [11, Theorem 1.1], which have their roots in [16]. Precisly, we argue the mathematical induction on the degrees of the polynomials P₁ and P₂.

If d₁ = d₂ = 0, then by Lemma 2.4, we directly attain

for all p ≥ 2. Also, if d₁ = 0 or d₂ = 0, then by [17, Theorem 1.1], it is easy to satisfy the inequality (1.3) for all p ≥ 2.

Now, assume that (1.3) is true for any polynomial P₁ of degree less than or equal to d₁ and for any polynomial P₂ of degree d₂. We need to show that (1.3) is still true if degree(P₁) = d₁ + 1, and degree(P₂) = d₂. Without loss of generality, we may assume P₁(x) = ∑|α|≤d1+1 a_α x^α is a polynomial of degree d₁ + 1 such that ∑α=d1+1 a_α∥ = 1 and does not contain ∥x∥^d₁+1 as one of its terms. Also, we may assume P₂(y) = ∑|β|≤d2 b_βy^β is a given polynomial of degree d₂ such that ∑β=d2 ∥b_β∥ = 1 and does not contain ∥y∥^d₂ as one of its terms. By duality and a simple change of variables, we have

where

Choose two collections of 𝓒^∞ functions {Υ_i}_i∈Z and {Γ_j}_j∈Z on (0, ∞), that satisfying the following conditions:

Define the multiplier operators S_j,i in Rⁿ⁺¹ × R^m+1 by

Set

Thanks to Minkowski’s inequality, we have

where

and

Let us first estimate the L^p-norm of MΩ,ϕ,ψ,∞,∞P1,P2 (f). Define

Hence, by generalized Minkowski’s inequality, it is easy to reach

If p = 2, then by a simple change of variables, Plancherel’s theorem, Fubini’s theorem, and Lemma 2.3, we get that

However, if p > 2, then by the duality, there exists b ∈ L^(p/2)′(Rⁿ⁺¹ × Rⁿ⁺¹) with ∥b∥_{L^(p/2)′(Rⁿ⁺¹×R^m+1)} = 1 such that

So, by Hölder’s inequality and Lemma 2.2, we conclude that

where b͠(z, w) = b(–z, –w). Thus,

which when Combined with (3.4) gives that there is ϵ ∈ (0, 1) so that

for all p ≥ 2. Therefore, by (3.3) and (3.5), we obtain

for all p ≥ 2. Now, let us estimate the L^p-norm of MΩ,ϕ,ψ,0,0P1,P2 (f). Take Q₁(x) = ∑|α|≤d1 a_α x^α, and define MΩ,ϕ,ψ,0,0Q1,P2 (f) and MΩ,ϕ,ψ,0,0P1,P2,Q (f) by

and