On the Hilbert Transform and C Families of Lines
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چکیده
We study the operator Hvf(x) := p.v. ∫ 1 −1 f(x− yv(x)) dy y defined for smooth functions on the plane and measurable vector fields v from the plane into the unit circle. We prove that if v has 1+2 derivatives, then Hv extends to a bounded map from L(R) into itself. What is noteworthy is that this result holds in the absence of some additional geometric condition imposed upon v, and that the smoothness condition is nearly optimal. Whereas Hv is a Radon transform, for which there is an extensive theory, see e.g. [5], our methods of proof are necessarily those associated to Carleson’s theorem on Fourier series [3], and the proof given by Lacey and Thiele [10]. A previous paper of the authors [9], has shown how to adapt these ideas to Hv; herein these ideas are combined with a crucial maximal function estimate that is particular to the smooth vector field in question.
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تاریخ انتشار 1999