The maximal Hilbert transform along nonconvex curves

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.

For , let be a curve in with . To Γ we associate the Hilbert transform which is defined as a principal value integral

where and . Similarly, one can define the corresponding maximal function and the maximal Hilbert transform as

and

The -boundedness for the Hilbert transform and the maximal function above have been well studied by many scholars. See [10] 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 [10]).

Theorem 1.1

(A) If Γ is a two-sided homogeneous curve in, then

(B) Assume that for smallt, lies in the subspace spanned by. Then the maximal Hilbert transform

is bounded fromto itself, .

For and , Córdoba and Rubio de Francia considered the case and , with the following properties:

(i) γ is biconvex, i.e., is decreasing in and increasing in ;

(ii) has doubling time, i.e., there exists a constant such that ;

(iii) γ is balanced, by which we mean the following: there exists such that for every .

They proved the following theorem in [7].

Theorem 1.2Under the assumptions (i), (ii) and (iii) on theγ, the maximal Hilbert transformis a bounded operator infor.

In this note, we consider the curve Γ with the form , where is a real-valued polynomial of t in , γ satisfies

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 [1].

Theorem 1.4Suppose thatPis a polynomial, γsatisfies (1.1) and. If, , and either (1) is zero, or (2) is nonzero and, then

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 [5] and [9].

Motivated by Bez’s result above, we obtain the -boundedness for . More precisely, we prove the following theorem.

Theorem 1.6Suppose thatPis a polynomial, γsatisfies (1.1) and. If, and either (1) is zero, or (2) is nonzero and, then

Moreover, the constantCdepends only onp, γand the degree ofP.

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 [1] and [3]. Notice that is a nonlinear operator, Minkowski’s inequality cannot be used as in Section 1 of [1], the linearization method is invoked to treat it. Similarly, the essential Proposition 1.2 in [1] 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.

Without loss of generality, we suppose that , where . Let be d-complex roots of P ordered as

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

where is the inverse image of a subset I restricted in .

The properties of P on and are important for our proof. The following related lemma can be found in [1] and [3].

Lemma 2.1There exists a numbersuch that for anyand any,

(i) for;

(ii) for, for;

(iii) for;

(iv) andfor, .

The following fact can be induced from the proof of Lemma 2.1 (see [1]), that is, we can choose such that, for ,

Let λ be the doubling constant for , define . Let I be a subset of , and , and are given by

and

For and , we define cones in by

and the corresponding projection operators by . Then, we have the following lemma which is Lemma 1 in [7].

Lemma 2.2Forand, we have

(i) ;

(ii) ;

(iii) .

The bootstrap argument (see [8]) plays an important role in the proof of the main result, so we present the following well-known result which can be found in [2] and [7].

Lemma 2.3Suppose thatis a sequence of positive operators uniformly bounded in, andis bounded infor, then

for.

Let and be given as in Section 2, then

Note that and are finite sets, it suffices to show that and are -bounded, respectively.

3.1 The -boundedness for

Let be some measurable function from to such that

By Minkowski’s inequality, we can control the -norm of by

Let for some and , then

and

Notice that γ is convex and , so for . Thus,

where is the inverse function of .

(3.1), (3.2) and (3.3) yield

3.2 The -boundedness for

For and , set and define measures by

for . For any , there exists such that . Then

Therefore,

By the -boundedness of (see [1]), 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

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 [1]. 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 [1].

Lemma 3.1For all, the function

is singled-signed on.

Proposition 3.2Forand, if, then

Proof For fixed , let , then

Case 1. . If and , for , (2.1) implies

Note that is monotone on , this fact follows from Lemma 3.1. By Van der Corput’s lemma and (3.4), we get .

If γ is even, then . If γ is odd, Lemma 3.1 still holds for , can be considered in the same way. Then, we have

If and . In the same way, for , we have

In the same way as above, we can get

Case 2. . If and , (3.4) still holds for . By integrating by parts,

Essentially, we just need to consider the second term, which can be dominated by

In order to estimate the term , we define , then . By (3.4), for , it is obvious that

On the other hand, for ,

Further, by (2.1), for ,

Thus, combining (3.7), (3.9) and (3.8), we have

For , by (3.4),

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 .

If and , (3.6) holds for . The same arguments used above imply

where and are as previous ones. For above, we have

Thus, (3.9) and (3.12) give

For , by (3.12) and (2.1),

can be treated in the same way as in Case 1. Thus, (3.10), (3.11), (3.13) and (3.14) imply

 □

3.2.2 -estimates for

The case of even γ By a linear transformation, we have

Note that , then for any ,

where is the Hardy-Littlewood maximal function acting on in the first variable and is given by

Thus, we obtain

If we can show that is -bounded, according to the -boundedness of (see [1]), we will get

So, it suffices to prove the following result.

Lemma 3.3For, 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 .

Suppose that is increasing on , then

For , if is increasing on , then, for , is nonnegative and decreasing on . Furthermore, note that

Therefore, for , we have

where M is the Hardy-Littlewood maximal function.

If is decreasing on , write

where denotes the reflection of g. Notice that is nonnegative and decreasing on and . Similarly,

For , and are increasing on and , respectively. Then, is increasing on , that is, . According to (2.1), ; furthermore, for . Therefore, combining the convexity of γ, we have

where .

By the -boundedness of M, we complete the proof of Lemma 3.3. □

The case of odd γ We decompose as , where . Therefore,

For , note that

For , , we have

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

By Lemma 3.3 and the -boundedness for , we just need to prove that

To obtain (3.18), we denote the set by and define projection operators by . Then, we use the following majorization:

where .

We will prove that for and ,

and

(3.19) and interpolation between (3.20) and (3.21) give (3.18).

To prove (3.20), we use the fact , Lemma 3.3 and Lemma 2.3,

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

Notice that . (3.23) and (3.24) imply

For , by (3.25) and the convexity of γ, we have

and

This ends the proof of (3.22).

3.2.3 -estimates for

For fixed , , where are positive integers less than 5. Then

Therefore,

For operators , by Lemma 2.2 and Lemma 2.3, we have

where we have used the fact that .

Finally, Lemma 2.2, (3.26) and (3.27) give

3.2.4 -estimates for

The last term can be decomposed as

where

Set , then, . By the -boundedness of and Lemma 2.2, we obtain

On the other hand, for , Plancherel theorem and Proposition 3.2 imply

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).

1. Bez, N: -boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves. Proc. Am. Math. Soc.. 135, 151–161 (2007)

2. Carbery, A, Christ, M, Vance, J, Wainger, S, Watson, D: Operators associated to flat plane curves: estimates via dilation methods. Duke Math. J.. 59, 675–700 (1989). Publisher Full Text

3. Carbery, A, Ricci, F, Wright, J: Maximal functions and Hilbert transforms associated to polynomials. Rev. Mat. Iberoam.. 14, 117–144 (1998)

4. Carbery, A, Vance, J, Wainger, S, Watson, D: The Hilbert transform and maximal function along flat curves, dilations, and differential equations. Am. J. Math.. 116, 1203–1239 (1994). Publisher Full Text

5. Carleson, H, Christ, M, Córdoba, A, Duoandikoetxea, J, Rubio de Francia, JL, Vance, J, Wainger, S, Weinberg, D: estimates for maximal functions and Hilbert transforms along flat convex curves in . Bull. Am. Math. Soc.. 14, 263–267 (1986). Publisher Full Text

6. Córdoba, A, Nagel, A, Vance, J, Wainger, S, Weinberg, D: bounds for Hilbert transforms along convex curves. Invent. Math.. 83, 59–71 (1986). Publisher Full Text

7. Córdoba, A, Rubio de Francia, JL: Estimates for Wainger’s singular integrals along curves. Rev. Mat. Iberoam.. 2, 105–117 (1986)

8. Nagel, A, Stein, EM, Wainger, S: Differentiation in lacunary directions. Proc. Natl. Acad. Sci. USA. 75, 1060–1062 (1978). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

9. Nagel, A, Vance, J, Wainger, S, Weinberg, D: Hilbert transforms for convex curves. Duke Math. J.. 50, 735–744 (1983). Publisher Full Text

10. Stein, EM, Wainger, S: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc.. 84, 1239–1295 (1978). Publisher Full Text