On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

Details On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

2007-04-05

arxiv.org/abs/0704.0808v3

Mathematics

Classical Analysis and ODEs

92 pages, 20+ figures. Final version of the paper. To appear in Memoirs AMS

Scientific paper

Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon f(x-yv(x)) \frac{dy}y where $ \epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ \epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1

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