Hilbert transform along a C^1+e vector field

Xiaochun Li

Let v be a vector field from R² to the unit circle S¹. We study the operator

Hvf(x) = p.v. ó õ 1 -1  f(x-tv(x))  dt t  .

We prove that if the vector field v has 1+e derivatives, then H_v extends to a bounded map from L² onto itself.