Boundedness of a class of rough maximal functions

In this work, we obtain appropriate sharp bounds for a certain class of maximal operators along surfaces of revolution with kernels in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}(\mathbf{S}^{n-1})$\end{document}Lq(Sn−1), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q > 1$\end{document}q>1. By using these bounds and using an extrapolation argument, we establish the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}Lp boundedness of the maximal operators when their kernels are in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L(\log L)^{\alpha}(\mathbf{S}^{n-1})$\end{document}L(logL)α(Sn−1) or in the block space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{0,\alpha-1}_{q} (\mathbf{S}^{n-1})$\end{document}Bq0,α−1(Sn−1). Our main results represent significant improvements as well as natural extensions of what was known previously.


Introduction and main results
Throughout this article, let R n , n ≥ 2, be the n-dimensional Euclidean space and 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 ∈ R n \ {0} and p denote the exponent conjugate to p; that is, 1/p + 1/p = 1.
Let K Ω,h (y) = Ω(y)h(|y|)|y| -n , where h : [0, ∞) → C is a measurable function and Ω is a homogeneous function of degree zero on R n that is integrable on S n-1 and satisfies the cancelation property S n-1 Ω x dσ x = 0.
We point out that the study the maximal operator M (γ ) P,Ω,φ was initiated by Al-Salman in his work in [11]. In fact, he investigated the L p (p ≥ 2) boundedness of M (2) P,Ω,t under the condition Ω ∈ L(log L) 1/2 (S n-1 ) ∪ B (0,-1/2) q (S n-1 ) for some q > 1. For more information about the importance and the recent advances on the study of such operators, the readers are referred to [1,2,5,27], and the references therein.
In view of the results in [4] as well as the results in [11], it is natural to ask whether the parametric maximal operator M (γ ) P,Ω,φ is bounded on L p (R n+1 ) under weak conditions on Ω, φ, and γ . We shall obtain an answer to this question in the affirmative as described in the next theorem. Precisely, we will establish the following result. Theorem 1.1 Suppose that Ω ∈ L q (S n-1 ), q > 1, and satisfy condition (1.1) with Ω L 1 (S n-1 ) ≤ 1. Suppose also that φ : R + → R is in C 2 ([0, ∞)), convex and increasing function with φ(0) = 0. Let P : R n → R be a polynomial of degree m and M (γ ) P,Ω,φ be given by (1.2). Then there exists a constant C p,q > 0 such that for γ ≤ p < ∞ and 1 < γ ≤ 2; and

4)
where β Ω = log(e + Ω L q (S n-1 ) ), C p,q = 2 1/q 2 1/q -1 C p , and C p is a positive constant that may depend on the degree of the polynomial P but it is independent of Ω, φ, q, and the coefficients of the polynomial P.
Here and henceforth, the letter C denotes a bounded positive constant that may vary at each occurrence but is independent of the essential variables.

Preliminary lemmas
This section is devoted to present and prove some auxiliary lemmas which will be used in the proof of Theorem 1.1. We start with the following lemma which can be derived by applying the arguments (with only minor modifications) used in [11].
is an arbitrary function on R + , and assume also that P = |α|≤m a α x α is a polynomial of degree m ≥ 1 such that |x| m is not one of its terms and |α|=m |a α | = 1.
Then a positive constant C exists such that Proof On the one hand, it is clear that Also, it is easy to get that Without loss of generality, we may assume that m > 1. Then, we follow the same steps as in [11, (2.9)-(2.12)] to prove that the inequality We shall need the following lemma which can be acquired by using the argument employed in the proof of [14,Lemma 4.7].

Lemma 2.2
Let Ω ∈ L 1 (S n-1 ) be a homogeneous function of degree zero and satisfy condi- Then, for 1 < p ≤ ∞, there exists a positive number C p so that Using a similar argument as in the proof of [4, Theorem 1.6], we obtain the following.
Proof Since L q (S n-1 ) ⊆ L 2 (S n-1 ) for q ≥ 2, it is enough to prove this lemma for 1 < q ≤ 2. It is clear that More precisely, we require the following: Define the multiplier operators S k in R n+1 by Hence, for f ∈ S(R n+1 ), we have By using [4, ineq. (3.10)] together with Lemma 2.2, we get for some constant 0 < ε p < 1 and for all 2 ≤ p < ∞. Therefore, by (2.6) and (2.7), we immediately satisfy inequality (2.5) for all 2 ≤ p < ∞.

Proof of the main results
Proof of Theorem where N : L p (R n+1 ) → L p (L γ (R + , dr r ), R n+1 ) is a linear operator defined by Now if we assume that for 2 ≤ p < ∞; and then by applying the interpolation theorem for the Lebesgue mixed normed spaces to the last two inequalities, we directly obtain for γ ≤ p < ∞ with 1 < γ ≤ 2; and M (1) . Thus, to prove our theorem, it is enough to prove it only for the cases γ = 1 and γ = 2.
Case 1 (if γ = 1). Assume that h ∈ L 1 (R + , dr r ) and f ∈ L ∞ (R n+1 ). Then, for all (x, Hence, by taking the supremum on both sides over all h with h L 1 (R + , dr r ) ≤ 1, we reach for almost every where (x, x n+1 ) ∈ R n+1 , which implies Case 2 (if γ = 2). We use the induction on the degree of the polynomial P. If the degree of P is 0, then by Lemma 2.3 we get that, for all p ≥ 2, Now, assume that (1.3) is satisfied for any polynomial of degree less than or equal to m with m ≥ 1. We need to show that (1.3) is still true if deg(P) = m + 1. Let a γ x γ be a polynomial of degree m + 1. Without loss of generality, we may assume that |γ |=m+1 |a γ | = 1, and also we may assume that P does not contain |x| m+1 as one of its terms. Let {ϕ k } k∈Z be a collection of C ∞ (0, ∞) functions satisfying the following conditions: Define the multiplier operators S k in R n+1 by and set Thanks to Minkowski's inequality, we have where Hence, by generalized Minkowski's inequality, it is easy to show that If p = 2, then by a simple change of variables, Plancherel's theorem, Fubini's theorem, and Lemma 2.1, we get that However, if p > 2, then by the duality, there exists Ψ ∈ L (p/2) (R n+1 ) with Ψ L (p/2) (R n+1 ) = 1 such that So, by Hölder's inequality and Lemma 2.2, we conclude that where Ψ (z, z n+1 ) = Ψ (-z, -z n+1 ). Thus, which when combined with (3.6) gives that there is 0 < ν < 1 so that for all p ≥ 2. Therefore, by (3.5) and (3.7), we obtain Thus, by Minkowski's inequality, we deduce On the one hand, since deg(Q) ≤ m, then by our assumption, for all p ≥ 2. On the other hand, since we have then by the Cauchy-Schwarz inequality, we reach that Hence, by Lemma 2.2, we get that for all p ≥ 2. Therefore, by (3.9)-(3.11), we obtain Consequently, by (3.4), (3.8), and (3.12), we finish the proof of Theorem 1.1.

Further results
In this section, we present some additional results that follow by applying Theorems 1.1 and 1.2. The first result concerns the boundedness of oscillatory singular integrals. More precisely, we deduce the following.
The generalized parametric Marcinkiewicz operator related to the operator M for 1 ≤ γ ≤ 2, it is easy to derive the following result.
We point out that by specializing to the case P = 0, γ = 2 and φ(t) = t, then the operator μ (γ ) P,Ω,φ (denoted by μ Ω ) is just the classical Marcinkiewicz integral operator introduced by Stain in [29] in which he showed that μ Ω is of type (p, p) for 1 < p ≤ 2 provided that Ω ∈ Lip α (S n-1 ) for some 0 < α ≤ 2. Subsequently, the operator μ Ω has been studied by many authors (for instance, see [11,13,15,18], as well as [19] and the references therein). For the significance and recent advances on the study of the generalized parametric Marcinkiewicz operators, we refer the readers to consult [7] and [6] among others.