Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability
In this note we establish certain weighted estimates for a class of maximal functions with rough kernels along “polynomial curves” on . As applications, we obtain the bounds of the above operators on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces, and on the vector-valued function spaces. Particularly, the above bounds are independent of the coefficients of the polynomials in the definition of the operators.
During the last several years, a considerable amount of attention has been given to the investigation of the boundedness for various kinds of integral operators on the Lebesgue spaces and other more general function spaces (see [1–8], for examples). The primary aim of this article is to establish the boundedness for maximal operators related to singular integrals on the mixed radial-angular spaces.
Let be the -dimensional Euclidean space and denote the unit sphere in equipped with the induced Lebesgue measure . Assume that is a homogeneous function of degree zero and satisfiesLet be a real polynomial on of degree satisfying and be a suitable function defined on ; we define the singular integral operator along the “polynomial curve” on bywhere for . Let and denote the set of all measurable functions satisfyingLet be defined as in (2) and let denote the closed unit ball in . Define the maximal function by
If and , we denote and .
As a formal extension of the Lebesgue space , the mixed radial-angular space has already been successfully used in studying Strichartz estimates and dispersive equations (see [9–16]) over the last several years. Recall that the mixed radial-angular spaces , , consist of all functions satisfying , where and It is clear that the spaces have the following easy properties.
(i) If and , then
(ii) If is a radial function on and , , then
(iii) If and , then Here the notation means that there are two positive constants such that and . Throughout this paper, we use to denote positive constants that depend on parameters .
Recently, the mixed radial-angular space is also playing active roles in the theory of singular integral operator (see [17–19]). In , among other things, Córdoba proved that is bounded on for all and , provided that . Later on, D’Ancona and Lucà  extended the above index to the range by applying the same argument in [18, Theorem 2.1] (see also [17, Theorem 1.1] for the weighted case). Recently, Liu and Fan  improved the above result to the case and extended the above results to the singular integral operators along polynomial curves. Precisely, let be the set of all measurable functions defined on satisfying Liu and Fan  proved the following result. Our main results can be formulated as follows.
Theorem A (see ). Let be a real polynomial on of degree and satisfy . Suppose that satisfies (1) and for some . Then for , the following inequalities hold: Here the above constants are independent of the coefficients of .
On the other hand, the classical maximal operator was originally introduced by Chen and Lin  who proved that if , then is of type for any and the range of is best possible. Subsequently, the mapping properties of have been discussed extensively by many authors (see [22–26], for example). Particularly, Al-Salman  proved the following result.
It is well known that the following are valid.
It follows from (14) and Theorem B that the condition implies the -boundedness of for . A question that arises naturally is the following.
Question A. Is the operator bounded on for if for some ?
In this paper we will give an affirmative answer to Question A. In order to obtain the boundedness for , we shall establish the following weighted estimates for .
Theorem 1. Let be a real polynomial on of degree and satisfy . Suppose that satisfies (1) and for some . Then, for any nonnegative measurable function on , it holds that provided that one of the following conditions holds:(i) and ;(ii) and . Here and , for any . is the Hardy-Littlewood maximal operator iterated times for all . Specially, when . is a maximal operator defined as in Section 2.
As applications of Theorem 1, we can obtain the following result.
Theorem 2. Let be the maximal operator defined by (4). Let be a real polynomial on of degree and satisfy . Suppose that satisfies (1) and for some . Then, for and or , the following inequalities hold:Here the constants are independent of the coefficients of .
Remark 3. We remark that our main results are new even in the special case .
The rest of this paper is organized as follows. In Section 2 we shall present some notations and auxiliary lemmas. The proofs of Theorems 1 and 2 will be given in Section 3. It should be pointed out that the main idea in the proof of Theorem 1 is a combination of ideas and arguments from [18–20, 27–29]. The proof of Theorem 2 is based on Theorem 1 and a criterion established in Section 2 (see Proposition 7). Throughout this note, for any , we let denote the dual exponent to defined as . We also use the convention . In what follows, for any function , we define by . For , we set
2. Preliminary Lemmas
In this section we shall give some notations and necessary lemmas, which will play key roles in the proof of our main results. In what follows, we assume that with . Let for and . For , we define a family of measures and the related maximal operator by where is defined in the same way as , but with replaced by . We also define the maximal operator by where is defined in the following way
Lemma 4 ([30, p.186, corollary]). Let and , where are real parameters and are distinct positive (not necessarily integer) exponents. Thenwith and does not depend on as long as .
Lemma 5. Let satisfy (1) and for some . Then for and , the following estimates hold:
Proof. Estimate (24) is trivial. By Lemma 4, Hölder’s inequality, and the changes of variable, we have for any . This together with (24) yields (25). Similarly, we can prove (26). It is clear that which together with (24) and the fact that yields (27)-(28) and completes the proof.
Lemma 6. Let satisfy (1) and for some . Then for , we havefor all .
Proof. We shall prove (31) by induction on . It is easy to see thatThis yields (31) for . Suppose that (31) holds for with . We shall prove (31) for . Let be a nonnegative Schwartz function supported in satisfying when . Define the measures by By Lemma 5 and the definition of , we havewhereBy (34) and Plancherel’s theorem, we getFrom (36)-(38) and our assumption, we haveBy the lemma on page 544 of  (, ), we haveFrom our assumption, (35) and (40), we getBy the lemma on page 544 of  (, ), we haveBy using this argument repeatedly, one can obtain ultimatelyCombining (43) with (35) and assumption yields thatThis completes the proof of Lemma 6.
To prove Theorem 1, we need the following proposition, which is of interest in its own right.
Proposition 7. Let and . Suppose that is a linear or sublinear operator such thatfor any nonnegative measurable function on , where is a bounded operator from to itself for all . Then for any , the following inequalities hold:
Proof. We only prove (46) since (47) and (48) can be proved similarly. The argument of the proof for (46) is similar to those of the proof in [19, Theorem 2.6] essentially. Fix and write . We can choose a number such that . Let denote the set of all with . By changes of variables, one hasFix . Let and . By (45), Hölder’s inequality, and changes of variables, which together with (49) leads to (46).
We begin with the proof of Theorem 1.
Proof of Theorem 1. By the duality we can writeLet be a nonnegative Schwartz function supported in satisfying when . For , we define the Borel measures on by One can verify thatEquation (51) together with (53) and Minkowski’s inequality yields thatIt follows from  that and for . From (55) we haveBy (56) and (57), to prove Theorem 1, it suffices to show thatholds for all and any nonnegative measurable function on , provided that one of the following conditions holds:(i) and ;(ii) and .We now prove (58) for the case and . For , let such that and . Define the Fourier multiplier operators by , where . It was shown in  thatandfor all and .
By the changes of variables and Minkowski’s inequality, we can writeHence, by (61), to prove (58) for the case and , it suffices to show that for any and , there exists independent of such thatWe now prove (62). Fix a nonnegative measurable function on . By (54) and Plancherel’s theorem, we havefor arbitrary function on . One can easily check thatfor any . By (63)-(64) and the interpolation of -spaces with change of measure ([32, Theorem 5.4.1]), we obtainfor any . By (65) with , (60), and the well-known property of the Rademacher’s function, we obtain It follows thatfor any .
On the other hand, fix , and it is easy to see thatThe interpolation between (68) and (69) implies thatfor all . It follows from (70) thatfor all . On the other hand, we have