1,572 research outputs found
On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields
Let 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 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 is Lipschitz,
there is a positive for which the transform above is bounded on . Our principal result gives a sufficient condition in terms of the
boundedness of a maximal function associated to . This sufficient condition
is that this new maximal function be bounded on some , for some . We show that the maximal function is bounded from to weak for all Lipschitz maximal function. The relationship between our results
and other known sufficient conditions is explored.Comment: 92 pages, 20+ figures. Final version of the paper. To appear in
Memoirs AM
- …