Lp estimates for maximal functions along surfaces of revolution on product spaces

Abstract This paper is concerned with establishing Lp estimates for a class of maximal operators associated to surfaces of revolution with kernels in Lq(Sn−1 × Sm−1), q > 1. These estimates are used in extrapolation to obtain the Lp boundedness of the maximal operators and the related singular integral operators when their kernels are in the L(logL)κ(Sn−1 × Sm−1) or in the block space Bq0,κ−1 $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$(Sn−1 × Sm−1). Our results substantially improve and extend some known results.


Introduction and main results
Let n, m ≥ , and let R N (N = n or m) be the N-dimensional Euclidean space. Let S N− be the unit sphere in R N equipped with the normalized Lebesgue surface measure dσ = dσ(·). Also, let x = x/|x| for x ∈ R n \ { }, y = y/|y| for y ∈ R m \ { }.
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 S n− × S m− that satis es For suitable mappings φ, ψ : R + → R, consider the singular integral operator T P ,P Ω,h,φ,ψ de ned, initially for C ∞ functions on R n+ × R m+ , by where (x, y) = (x, x n+ , y, y m+ ) ∈ R n+ × R m+ and P : R n → R, P : R m → R are two real-valued polynomials.
When P (u) = and P (v) = , we denote T P ,P Ω,h,φ,ψ 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 Fe erman in [1] in which he obtained the L p boundedness of T Ω,h for all < p < ∞ whenever Ω satis es some regularity conditions and h ≡ . As a matter of fact, the systematic study of such operator began by Fe erman in [1], and then it was elaborated very much by Fe erman 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 (R n × R m ) for < p < ∞ when Ω ∈ L(log L) (S n− × S m− ) and h satis es certain integrability-size condition. Furthermore, the authors of [3] established the optimality of the condition in the sense that the space L(log L) (S n− × S m− ) cannot be replaced by L(log L) −ε (S n− × S m− ) for any < ε < . 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 C ([ , ∞)), convex and increasing functions with φ( ) = ψ( ) = , then Al-Salman in [4] showed that T Ω, ,φ,ψ is bounded on Recently, Al-Salman improved this result in [6]. In fact, when φ, ψ are given as in [4], he veri ed the L p boundedness of T Ω,h,φ,ψ for all p ∈ ( , ∞) under the conditions Ω ∈ L(log L)(S n− × S m− ) and h ∈ L (R + × R + , drdt rt ) with h L (R + ×R + , drdt rt ) ≤ . The maximal operator that related to our singular integral operator is M P ,P Ω,φ,ψ that given by Again, when P (u) = and P (v) = , we denote M P ,P Ω,φ,ψ 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 L boundedness of M Ω whenever Ω ∈ L(log L) (S n− ×S m− ). 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 ≥ provided that Ω ∈ L(log L)(S n− × S m− ). Moreover, they pointed out that the condition Ω ∈ L(log L)(S n− × S m− ) is optimal in the sense that the exponent in L(log L)(S n− × S m− ) cannot be replaced by any smaller positive number τ < so that M Ω is bounded on L (R n+ × R m+ ). Also, an improvement of the result in [7] was obtained by Al-Qassem in [10]. Indeed, Al-Qassem established the L p ( ≤ p < ∞) estimates for the class M Ω whenever Ω belongs to the block space B ( , ) q (S n− × S m− ) for some q > . Furthermore, he proved that the condition Ω ∈ B ( , ) q (S n− × S m− ) is nearly optimal in the sense that the operator M Ω may lose the L boundedness if Ω is assumed to be in the space B ( ,ε) q (S n− × S m− ) for some − < ε < . Recently, it was found in [6] that the maximal operator M Ω,φ,ψ is bounded on L p (R n+ × R m+ ) for any p ≥ if Ω ∈ L(log L)(S n− × S m− ), and φ, ψ are in C ([ , ∞)), convex and increasing functions with φ( ) = ψ( ) = . Very recently, when φ(t) = ψ(t) = t, Al-Dolat and et al. found in [11] that the L p (p ≥ ) boundedness of M P ,P Ω,φ,ψ is obtained under the condition Subsequently, the investigation of the L p boundedness of M P ,P Ω,φ,ψ under weak conditions has received much attentions from many mathematicians. For the signi cance of considering the integral operators M P ,P Ω,φ,ψ , we refear the readers to consult [8] and [11][12][13], among others.
The main result of this work is formulated as follows: for all p ≥ , where Cp,q = /q /q − Cp and Cp is a positive constant that may depend on the degrees of the polynomials P , P but it is independent on Ω, φ, ψ, q, and the coe cients of the polynomials P , P .
We remark that by the result in Theorem 1.1 and using an extrapolation argument, we get that M P ,P Ω,φ,ψ is bounded on Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.

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

Lemma 2.1.
Let Ω ∈ L q (S N− ), q > be a homogeneous function of degree zero on R N with Ω L (S N− ) ≤ , and let φ : R + → R be a C ([ , ∞)), convex and increasing function with φ( ) = . Consider the maximal function N Ω,φ given by Then for p > and f ∈ L p (R N+ ) there exists a positive number Cp such that Then for all f ∈ L p (R n+ × R m+ ) and p > , the maximal function It is easy to prove the above lemma by using Lemma 2.1 and the inequality , and • denotes the composition of operators. A signi cant step toward proving Theorem 1.1 is to estimate the following Fourier transform: . Assume that φ, ψ are arbitrary functions on R + , and assume also that P = |α|≤d aαx α is a polynomial of degree d ≥ such that |x| d is not one of its terms and |α|=d |aα| = ; Then, a positive constant C exists such that Proof. On one hand, it is trivial to get that Also, it is easy to see that Without lossing of generality, we may assume that d > . Hence, by Van der-Corput Lemma, we obtain Combine the last inequality with the trivial estimates for any < θ < . In the same manner, we derive Thus, using Hölder's inequality leads to We shall need the following Lemma which can be acquired by using the arguments employed in the proof of [ Proof. Choose collections of functions {Φ i } i∈Z and Ψ j j∈Z de ned on R n and R m , respectively with the following properties: De ne the multiplier operators S j,i in R n+ × R m+ via the Fourier transform given by Hence, for any f ∈ C ∞ (R n+ × R m+ ), we have Therefore, by using [6, Theorem 4.1], we get for some constants < ε , ε < and for all ≤ p < ∞. Consequently, the inequality ( . ) follows by using ( . ) and ( . ).

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 P and P . Now, assume that ( . ) is true for any polynomial P of degree less than or equal to d and for any polynomial P of degree d . We need to show that ( . ) is still true if degree(P ) = d + , and degree(P ) = d . Without loss of generality, we may assume P (x) = |α|≤d + aαx α is a polynomial of degree d + such that |α|=d + |aα| = and does not contain |x| d + as one of its terms. Also, we may assume P (y) = |β|≤d b β y β is a given polynomial of degree d such that |β|=d b β = and does not contain |y| d as one of its terms. By duality and a simple change of variables, we have Choose two collections of C ∞ functions {Υ i } i∈Z and Γ j j∈Z on ( , ∞), that satisfying the following conditions: De ne the multiplier operators S j,i in R n+ × R m+ by Thanks to Minkowski's inequality, we have Hence, by generalized Minkowski's inequality, it is easy to reach If p = , then by a simple change of variables, Plancherel's theorem, Fubini's theorem, and Lemma 2.3, we get that However, if p > , then by the duality, there exists So, by Hölder's inequality and Lemma 2.2, we conclude that which when Combined with ( . ) gives that there is ∈ ( , ) so that for all p ≥ . Therefore, by ( . ) and ( . ), we obtain Thus, by Minkowski's inequality, we deduce On one hand, since deg(Q ) ≤ d , then by induction step we have for all p ≥ . On the other hand, it is easy to check that e iP (ru) − e iQ (ru) ≤ r (d + ) |α|=d + aαu α ≤ r d + .
So, by following a similar argument as in [18] and by Cauchy-Schwartz inequality, we have that where • denotes the composition of operators, N Ω,ψ f (x, y) = N Ω,ψ f (·, y)(x) is the maximal function de ned as in Lemma 2.1; and M ( ) P ,Ωn ,φ (fr(x, y) = M ( ) P ,Ωn ,φ (fr(x, ·)(y) is the maximal operator in the one parameter setting de ned as in [17,Eq. (1.2)]. Hence, by following a similar argument as in [18, p. 607] together with [17] and Lemma 2.1, we get for all p ≥ . Therefore, by ( . )-( . ), we obtain that for all p ≥ , In the same manner, we can derive that and for all p ≥ . Consequently, by ( . ), ( . ) and ( . )-( . ), we satisfy the inequality ( . ) for any polynomial P of degree d + and for any polynomial P of degree d . Similarly, we can show that the inequality ( . ) holds for any polynomial P of degree d + and for any polynomial P of degree d . This completes the proof of Theorem 1.1.
The purpose of this section is to study the L p boundedness of the singular inegral operator T P ,P Ω,h,φ,ψ (f )(x, y) and the maximal operator M P ,P ,(γ) under weaker conditions, where M P ,P ,(γ) The rst result of this section is the following: for all γ ≤ p < ∞ with < γ ≤ ; and Proof. It is clear that if γ = , then we have M P ,P ,(γ) Ω,φ,ψ = M P ,P Ω,φ,ψ . So, by Theorem 1.1, the inequality ( . ) holds for all p ≥ . However, if γ = ; we assume that h ∈ L (R + × R + , drdt rt ) and f ∈ L ∞ (R n+ × R m+ ). Then for all (x, y) ∈ R n+ × R m+ , we have Hence, by taking the supremum on both sides over all h with h ≤ , we reach for almost every where (x, y) ∈ R n+ × R m+ , which leads to Finally, if < γ ≤ . We follow a similar approach as in [15]. By duality, we get ,Ω (f ) L p (L γ (R + ×R + , drdt rt ),R n+ ×R m+ ) . It is worth mentioning that when φ(t) = ψ(t) = t and P (u) = P (v) = , Al-Qassem and Pan in [8] extended the results of Theorem 4.1. In fact, they established the L p boundedness of M P ,P ,(γ) By the conclusion in Theorem 4.1 and applying an extrapolation argument (see [16,19,20]), we shall improve and extend the corresponding results in [4,6,8,11,13]. Precisely, we obtain the following: Theorem 4.2. Suppose that P , P , φ, and ψ are given as in Theorem 1.1 Proof. The idea of proving Theorem 4.2 is taken form [17], which has its roots in [16] as well as in [19]. When Ω ∈ L(log L) /γ (S n− × S m− ) with < γ ≤ and Ω satis es the conditions ( . )-( . ), then Ω can be decomposed as a sum of functions in L (S n− × S m− ) (see [21]). In fact, we have Hence, it is easy to see that and + log /γ (e + Ω k ∞ ) ≤ + log /γ (e + C k ) ≤ Ck /γ . (4.8) As Ω ∈ L (S n− × S m− ), then by Thorem 4.1 we get for γ ≤ p < ∞. Therefore, by Minkoswski's inequality and ( . )-( . ), we obtain that However, when Ω ∈ B ( , /γ − ) q (S n− × S m− ) with q > , < γ ≤ and Ω satis es the conditions ( . )-( . ), then Ω can be written as where each cµ is a complex number, each bµ is a q-block supported in an interval Iµ on (S n− × S m− ) and For each µ, de ne the blocklike functionbµ bỹ It is clear that eachbµ (x, y) satis es the following: We point out that under the assumptions Ω belongs to the block space B ( , ) q (S n− × S m− ), h ∈ ∆γ R + × R + for some q, γ > , and when φ, ψ are C ([ , ∞)), convex increasing functions with φ( ) = ψ( ) = , the author of [22] proved that for every p satisfying | /p − / | < min / , /γ , there exists a constant Cp such that T Ω,h,φ,ψ (f ) p ≤ Cp f p for every f ∈ L p R n+ × R m+ . By this result, it is clear that the range of p is the full range ( , ∞) whenever h ∈ L γ (R + × R + ) with γ ≥ . But what is about the L p boundedness of T Ω,h,φ,ψ when h ∈ L γ (R + × R + ) for < γ < . We shall obtain an answer to this question in the a rmative as described in the following theorem. . Let h ∈ L γ (R + × R + ) for some < γ ≤ , and let φ, ψ be given as in Theorem 1.1. Then the singular integral operator T P ,P Ω,h,φ,ψ (f )(x, y) is bounded on L p (R n+ × R m+ ) for all < p < ∞.