Research Article | Open Access
Hilbert Transforms along Convex Curves for -Valued Functions
We show that Hilbert transforms along a large class of convex curves are bounded on , where , .
Let be a curve in with , . For , the Hilbert transform along curve is defined by the following principal-valued integral:
There has been considerable interest to determine for which curves , and which indices , one has for a constant depending only on and . This problem has been extensively studied by a large number of authors. More results are found in [1–6].
The question of whether these results could be extended to the Lebesgue-Bôchner spaces of vector-valued functions was taken up by several authors recently, where is some suitable Banach space. But the Banach space of most interest to us will be for . Let us state some previous theorems which establish the background for our current work. The first is the work done in 1986 by Rubio de Francia et al. . They dealt with well-curved curves and obtained the following result.
Theorem 1. Let be a well-curved curve in with . Then the -valued inequality holds for all , with , , and all , .
Recently, the author considered the submanifold of finite type which means that the image of the mapping is of finite type at . We first restrict our attention to that are given as polynomial functions, because it is a model problem in this situation. We obtained the following.
Theorem 2 (see [8, 9]). (1) Let be a polynomial function. Then the -valued inequality
holds for all , with , , and all , .
(2) Let the image of be of finite type at . Then the -valued inequality holds for all , with , , and all , .
An important feature of the well-curved curve is that it is in a sense approximated by the model curves; the models are homogeneous with respect to some nonisotropic dilations. This means that corresponding scaling arguments are essential for well-curved curves.
“The image of is of finite type at ” is a crucial condition imposed on , this condition bears on finitely many derivatives of at the origin, this property is captured by a Taylor polynomial of at origin of sufficiently high degree and should be modeled by polynomial behavior. For polynomial function , in order to utilize the scaling arguments, the situation need be simplified further. This is accomplished by using the lifting technique.
All these results have common roots in that both depend in a key way on the use of scaling arguments or nonisotropic dilations, although these dilations are given implicitly for above curves in different way. More precisely, an analytic family of convolution operators are defined such that . The result of is obtained with the help of the Fourier transform and Plancherel theorem for all with , where is some positive constant. is closer to a more traditional singular integral when , because there is some slack in the estimate of the kernel. Therefore, by the vector-valued Calderón-Zygmund theory of Benedek et al. in , one can obtain the result for all , with , , . By generalized analytic interpolation of operators (see Lemma 12), a combination of above two results gives the desired estimate for .
What about the situation when the curves do not have appropriate homogeneity as well-curved curves or finite type curves? This kind of curves has been extensively considered in [1–5]. The Calderón-Zygmund type argument cannot be used to show that is bounded in for , and . New techniques should be utilized. In this paper, we deal with plane curves of the form , where is a convex function for . We have the following main result.
Theorem 3. Suppose that is a continuous odd function, twice continuously differentiable, increasing and convex for . Suppose also that is monotone for and that there exists so that for . Then is bounded on for all , with , .
The proof of Theorem 3 is given in several steps, which are similar to that in . An analytic family of operators are introduced so that . The result is equivalent to the boundedness of the multiplier associated to with , where is some positive constant. Van der Corput lemma and integration by part are used to obtain the boundedness. The kernels of do not have appropriate homogeneity, we have to use the vector-valued Fourier multiplier theorem (See Lemma 11) to show that is bounded on for , with , , and , where is some negative constant. Finally, the theorem follows from above two facts by using generalized analytic interpolation of operators.
Remark 4. (1) Theorem 3 covers a large class of functions such as
The first one is homogeneous, while the other one is without homogeneity. We extend the class of curves for which the -result is known for , . On the other hand, we also generalized Theorem 3.1 of Nagel and Wainger in  to the -valued setting for .
(2) In the present work, the kernels must be smoothed a definite amount, we are unable to take or . Hence we obtain bounds for on only when , .
2. Proof of the Theorem
For , we define an analytic family of operators by where are given by Obviously, is our original operator .
2.1. The Boundedness of on
In this subsection, we prove that where for some .
Clearly, the boundedness of on is equivalent to the uniform boundedness of the corresponding multiplier . Thus, we just need to show that where the constant is independent of .
Lemma 5. Let , let , and suppose that has at most roots in . Then
The second one is Van der Corput lemma which plays an important role in estimating related multipliers. This lemma appears in several books or papers, compare, for example, Stein [11, P.332].
Lemma 6. Suppose is real-valued and smooth in , and that for all . Then holds when ; the bound is independent of and .
To estimate , for fixed , we choose so that and decompose as First of all, we consider the first integral. For any , Notice that for . Because of the convexity of , we have for . Thus, For the integral II, note that is odd. An easy calculation shows that It is trivial that is piecewise monotone in . Further, it has at most monotone intervals in . Indeed, the derivative of is is nonzero for . So, we consider the following equation: We suppose ; otherwise, it is obvious. Then, above equation is For and , we have . There are about periods in interval , only one root exists in one period. So, (19) has at most roots. That mean that has at most stationary points in , that is there are at most monotone intervals. The case can be treated in the same way. We suppose it is monotone, otherwise, we deal with the integral on every monotone interval. By the second mean value theorem and Lemma 5, there exists such that In the same way, we can prove that
We next deal with the the part of the integral where . If is monotone increasing, we set while if is monotone decreasing, put Obviously, . By integration by part, we have Notice that for or . Van der Corput lemma shows that in either case. A combination of with the hypothesis implies that . So, for the boundary terms, , we have Finally, we consider the integrated terms; it is bounded by The integral can be handled similarly. This completes the proof of (10).
2.2. The Boundedness of on
In this subsection, we show that where , , , the constant depends on , and is independent of .
To prove (28), we need a vector-valued Fourier multiplier theorem. We have to recall some definitions before we present the theorem.
Definition 7. Let be a Banach space; is said to be an UMD-space, if the Hilbert transform is bounded on for some (and then all) .
Definition 8. A basis in a Banach space is called unconditional, if there is a constant such that for every one has , for all for .
Definition 9. A Banach is said to have local unconditional structure if there exists a constant such that for any finite-dimensional subspace of there exists a Banach space with an unconditional basis and operators and such that the natural embedding admits a factorization and .
Remark 10. Well-known examples of -spaces are , and , , , three examples are also Banach lattices for their usual norm and the pointwise order. Further, all Banach lattices have local unconditional structure.
Let be a bounded function, we associate operators defined on the test functions by As a vector-valued Fourier multiplier theorem, we state the following vector-valued Mikhlin theorem which already was proved in .
Lemma 11. Let be an UMD-space with local unconditional structure. Then for any there is a constant such that
The uniform boundedness of is trivial, it can be established by minor modification of the proof of (10). Without repetition, we omit the proof.
2.2.1. The Boundedness of
Integration by part implies that
Note that , for , we have . The boundary terms is bounded by .
For , making the change of variables , we obtain
In the similar way, the second integrated term can be dominated by Therefore, for ,
2.2.2. The Boundedness of
Integrating by parts, we obtain
To estimate above two integrals, we follow the argument similar to that in the proof of (10). For the first integral, for any , it suffices to bound the following two parts Recall that was chosen so that , and because of the convexity. Thus, For , an elementary calculation shows that
Similarly, the second integral can be controlled by Therefore, for ,
2.2.3. The Boundedness of
To take care of , we note that it can be written as For the first term, integrating by parts, we obtain
Obviously, for , , . So, the boundary terms is bounded by .
For the first integrated terms, making the change of variables , one has
The second integrated terms can be treated in the same way, let ,
Similarly, a trivial calculation shows that The second term can be handled similarly. Integrating by parts, we decomposed it as
Obviously, for , , . The boundary term is dominated by .
For the first integrated terms, by making the change of variables , we have the estimate
To estimate the second integrated terms, we make the transformation and get
Similarly, the third integrated terms can be treated as Note that for , we have the following elementary estimates Finally, combining above eight estimates, we obtain
Let denote the closed strip . is a family of uniformly bounded linear operators on , that is, there is an such that Moreover, the function is continuous on and analytic in the interior of , whenever .
Lemma 12. With the above assumptions, if for , satisfies the following conditions:(i); (ii). Then we have where , and .
Let . Note that , then there exists a constant which is independent of such that Also, for , , there exists a constant which is independent of such that For , , there exists , so that where and satisfies the equation for some and . Therefore, by Lemma 12, we obtain