On the Hilbert transform and lacunary directions in the plane
Abstract
Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert transform replaced by the one dimensional maximal function was proved by Nagel, Stein and Wainger in 1978.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2000
- DOI:
- 10.48550/arXiv.math/0010196
- arXiv:
- arXiv:math/0010196
- Bibcode:
- 2000math.....10196L
- Keywords:
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- Classical Analysis and ODEs;
- 42b20
- E-Print:
- 11 pages, 1 fiure. This is version to appear in Illinois J. Math