The maximal Hilbert transform along nonconvex curves
© Liu; licensee Springer 2012
Received: 11 March 2012
Accepted: 17 August 2012
Published: 31 August 2012
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.
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 ).
- (A)If Γ is a two-sided homogeneous curve in , then
- (B)Assume that for small t, lies in the subspace spanned by . Then the maximal Hilbert transform
is bounded from to itself, .
γ is biconvex, i.e., is decreasing in and increasing in ;
has doubling time, i.e., there exists a constant such that ;
γ is balanced, by which we mean the following: there exists such that for every .
They proved the following theorem in .
Theorem 1.2 Under the assumptions (i), (ii) and (iii) on the γ, the maximal Hilbert transform is a bounded operator in for .
Definition 1.3 A function belongs to , if there exists a constant such that for . It is also said that f has doubling time.
For this case, Bez obtained the -boundedness of and in .
Moreover, the constant C depends only on p, γ and the degree of P.
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 .
Motivated by Bez’s result above, we obtain the -boundedness for . More precisely, we prove the following theorem.
Moreover, the constant C depends only on p, γ and the degree of P.
Remark 1.7 Let . Comparing those conditions for γ in Theorem 1.2, we find that conditions in Theorem 1.6 are stricter. But we should note that may be a nonconvex function.
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.
where is the inverse image of a subset I restricted in .
for , for ;
and for , .
and the corresponding projection operators by . Then, we have the following lemma which is Lemma 1 in .
3 Proof of Theorem 1.6
Note that and are finite sets, it suffices to show that and are -bounded, respectively.
3.1 The -boundedness for
where is the inverse function of .
3.2 The -boundedness for
where δ is the Dirac measure in , and is the identity operator. Then, we just need to estimate , and , respectively.
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.
3.2.1 Fourier transform estimates of
Before the proof of Proposition 3.2, we need the following lemma which is Lemma 2.2 in .
is singled-signed on .
Note that is monotone on , this fact follows from Lemma 3.1. By Van der Corput’s lemma and (3.4), we get .
Note that can be split into a finite number of disjoint intervals such that is singled-signed on each interval. Suppose that is such an interval and by (2.1), then . So .
3.2.2 -estimates for
So, it suffices to prove the following result.
Lemma 3.3 For , is a bounded operator in , .
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.
By the -boundedness of M, we complete the proof of Lemma 3.3. □
(3.19) and interpolation between (3.20) and (3.21) give (3.18).
This ends the proof of (3.22).
3.2.3 -estimates for
where we have used the fact that .
3.2.4 -estimates for
The author was supported by Doctor Foundation of Henan Polytechnic University (Grant: B2011-034).
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