The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain -boundedness for the maximal Hilbert transform associated to curves , where , P is a real polynomial and γ is convex on . Then, we can conclude that the Hilbert transform along curves exists in pointwise sense.
MSC: 42B20, 42B25.
Keywords:the maximal Hilbert transform; maximal functions; nonconvex curves; Littlewood-Paley theory
The -boundedness for the Hilbert transform and the maximal function above have been well studied by many scholars. See  for a survey of results through 1977. More recent results can be found in [1,2,4-7].
Appropriate estimates for the maximal Hilbert transform give the pointwise existence of the principal value Hilbert transform. So, we focus on the -bounds for the maximal Hilbert transform in this paper. Let us state some previous theorems to establish the background for our current work. The first result about is the work of Stein and Wainger (see ).
They proved the following theorem in .
For this case, Bez obtained the -boundedness of and in .
Moreover, the constantCdepends only onp, γand the degree ofP.
Remark 1.5 If , is “more convex” than in some sense, then Γ is “convex” enough for the -boundedness of and . In the case , the linear term of cannot improve the convexity of γ. To obtain the -boundedness for associated operators, one needs to pose additional condition(s) on γ, that is, . For more details, see  and .
Moreover, the constantCdepends only onp, γand the degree ofP.
The convexity of the polynomial P is important for our main result. P has different convexity in different intervals, which suggests that will be decomposed according to the properties of P. The decay of associated multipliers is essential for the proof of Theorem 1.6. This set of techniques originated from the work  and . Notice that is a nonlinear operator, Minkowski’s inequality cannot be used as in Section 1 of , the linearization method is invoked to treat it. Similarly, the essential Proposition 1.2 in  is useless for the maximal Hilbert transform. Littlewood-Paley theory and interpolation theorem are effective tools to treat those problems. Those ideas are due to the contribution of Córdoba, Nagel, Vance, Wainger, Rubio de Francia.
The organization of our paper is as follows. In Section 2, we list some key properties concerning the polynomial and give some lemmas for the proof of the main result. The -estimates for will be proved in Section 3.
Let A be a positive constant which will be chosen in Lemma 2.1. Define if it is nonempty for and . Let , then can be decomposed as , where is the interval between two adjacent . It is obvious that is disjoint. Then, we can decompose as
The following fact can be induced from the proof of Lemma 2.1 (see ), that is, we can choose such that, for ,
and the corresponding projection operators by . Then, we have the following lemma which is Lemma 1 in .
3 Proof of Theorem 1.6
(3.1), (3.2) and (3.3) yield
By the -boundedness of (see ), it suffices to consider the latter term. Let such that for and for . Write and denote by ⋆ convolution in the first variable. For , the truncated Hilbert transform can be decomposed as
The decay of is important for the boundedness of three maximal operators above. Essentially, estimates for in the following subsection have been proved in . We repeat them just for completeness.
Before the proof of Proposition 3.2, we need the following lemma which is Lemma 2.2 in .
In the same way as above, we can get
Essentially, we just need to consider the second term, which can be dominated by
Thus, combining (3.7), (3.9) and (3.8), we have
Thus, (3.9) and (3.12) give
The case of even γ By a linear transformation, we have
Thus, we obtain
If we can show that is -bounded, according to the -boundedness of (see ), we will get
So, it suffices to prove the following result.
Proof We denote by for short, then . Note that there is no root of in , that is, is singled-signed. For , , by (2) of Lemma 2.1, is also singled-signed on . By and the convexity of γ, for . Then, is monotonous on . On the other hand, by Lemma 3.1, for , is monotonous on .
where M is the Hardy-Littlewood maximal function.
Similar to the case of even γ, we obtain
For the second term in the right-hand side of (3.16),
Then, we get the estimate
(3.19) and interpolation between (3.20) and (3.21) give (3.18).
For (3.21), by Plancherel theorem,
So, it suffices to show
By (2.1), we have
On the other hand, by (2.1) and the convexity of γ,
By the argument similar to that in the proof of Proposition 3.2, we obtain
This ends the proof of (3.22).
Finally, Lemma 2.2, (3.26) and (3.27) give
The last term can be decomposed as
Interpolation between (3.29)-(3.30) and (3.28) gives
The authors declare that they have no competing interests.
The author was supported by Doctor Foundation of Henan Polytechnic University (Grant: B2011-034).
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