A Class of Singular Integral Operators Associated to Surfaces of Revolution *

for all (the Schwartz function class on 1 n f   1 n R  ), where   1 , n 1 n n x x R R R      . Operators of the type (1.2) have been studied quite extensively (see [1-13] and therein numerous references). We refer the readers to see Stein-Wainger’s report [14, 15] for more background information. In 1996, Kim, Wainger, Wright and Ziesler proved the following result. Theorem A [10]. Let be convex, increasing and


Let
, , be the n-dimensional Euclidean space and be the unit sphere in equipped with the normalized Lebesgue measure Let be the surface of revolution generated by a suitable function .For nonzero points , we denote ' .x x x  Let be a homogeneous function of degree zero on and satisfy Suppose that is a radial measurable function.Define the singular integral operator for all (the Schwartz function class on . Operators of the type (1.2) have been studied quite extensively (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and therein numerous references).We refer the readers to see Stein-Wainger's report [14,15] for more background information.In 1996, Kim, Wainger, Wright and Ziesler proved the following result.
Theorem A [10].Let be convex, in-creasing and is a homogeneous function of degree zero on and satisfies (1.1).Then ,1 By a minor modification of the proof in Theorem A, one can show that the conclusion of Theorem A remains valid if the condition is replaced by the condition for some (see [16, pp. 372-373], as well as [6]).Subsequently, this result was improved and extended by many authors (see [1,2,5,6,11,12] et al.).In particular, in 2001, Lu, Pan and Yang gave the general theorem as follows.
for some  and small , where is a constant independent of .Suppose that Actually, the condition can be weakened to the case: (see [9]), and the size condition on  in Theorem B is the best one, so far,  even if (see [8]).
On the other hand, for , it is known in [17], if , that , which is optimal and much weaker than that for the classical Calderon-Zygmund singular integral operators.It should be noted that the spaces and do not include in each other.
Inspired by Al-Salman's work [17], we shall establish the following main result in this paper. Theorem , , provided that the lower dimensional maximal operator is continuously differentiable on and satisfy for some  and small , where is a constant independent of , in particular, if , then the integral in (1.2) exists in principle-value sense when, say, , then M  is bounded on for , see [13].
, and there exists an is ether even or odd and there exists a so that for each then M  is bounded on for , see [3] or [4].
In order to obtain Theorem 1, we let , S   be the operator defined by  Therefore Theorem 1 can be deduced immediately from the next theorem.
Theorem 2. Let  and be as in Theorem 1.
By the similar arguments as in [1], we remark that the condition is optimal.Precisely, there exists an lies in for all . And it is worth pointing out that the size condition is much weaker than that for the classical Calderón-Zygmund singular integral operators.
This paper is organize as follows.In Section 2 we will give the proofs of our theorems.An extension of our main results will be given in Section 3. We would like to remark that the main ideas in the proofs of our results are taken from [7,9,17].
Throughout this paper, we always use letter to denote positive constants that may vary at each occurrence but are independent of the essential variables.

Proofs of Main Results
Let us begin with a lemma, which will play a key role in the proofs of our main results.
Lemma 2.1.Let and satisfy (1.1).If the maximal operator for, then the following maximal operator Thus by Minkowski's inequality, we have For each fixed This together with the p L -bondedness of M  , and change of variables, show that , where is independent of.Lemma 2.1 is proved.C Next we introduce some notations.Assume that and satisfies (1.1).For any Then we have the following: where Now we give the proofs of our theorems as follows.
Proof of Theorem 2. For each . By (2.5) and Minkowski's inequality, we have , let's argue as in the proof of Theorem 2.1 in [1], choose a collection of , , , , Then by (2.9) and Minkowski's inequality, in the following cases: Case 1.For , we claim that there exists where C is independent of l and k.Indeed, by Plancheral's theorem and Fubini's theorem, , dd , where In order to prove (2.12), we first estimate   , l j J  .Obviously, we have By (2.2), a straightforward computing shows that Using interpolation between (2.14) and (2.15), we get On the other hand, we have And by integration by parts, with the easy fact Thus, by Holder's inequality This together with (2.14) and an interpolation implies


(2.17) Then by the fact that , ,' where C is independent of l and k.Indeed, choose such that Applying Holder's inequality again, it follows from Lemma 2.1 and the Littlewood-Paley theory (see [18,Chapter 4]) that This together with (2.12) and an interpolation implies (2.18).Therefore, by (2.11), (2.12) and (2.6), we get for 1 , then by (2.11), (2.18) and (2.6), we obtain that for , which completes the proof of Theorem 2.
Proof of Theorem 1.By the fact and Theorem 2, it follows that ,h T  is bounded on for .On the other hand, by duality we can obtain that ,h T  1 p n L R 

Further Results
In this section, we will extend the definition of T ,h to higher dimensional cases.Let Φ(t)=( 1 (t),   In [10], Fan and Zheng extended the result of Theorem B to the operator T Φ,h for h∆ γ (R + ) for γ > 1.Here, we will obtain the following results.
Theorem 3. Suppose that hL 2 (R + , r −1 dr) and  L(log + L) 1/2 (S n−1 ).Then and C  is a constant independ- ent of l.For each j   and

4 .Lemma 4 . 1 .
provided that the maximal operator M  defined in (4.3)Let Φ and  be as in Theorem 3. Then 1)provided that the maximal operator M  defined in (4.3) m = 1 then Theorem 3 and 4 are reduced to Theorem 1 and 2. For m  2, by the same arguments as in the proof of Lemma 2.1, we can obtain the following lemma.Let L 1 (S n−1 ) and satisfy (1.1).If the maximal operator M Φ in (4.3) is bounded on L p (R m+1 ) for p > 1, then the following maximal operator M ,Φ defined by is bounded on for .Theorem 1 is proved.