Hilbert Transforms along Convex Curves for -Valued Functions

We show that Hilbert transforms along a large class of convex curves are bounded on , where , . 1. Introduction 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-B？chner 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. [7]. 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