Hilbert Transform on smooth families of lines

Michael Lacey
Georgia Institute of Technology

We consider a Radon transform of a particular type. Given a smooth vector field $v$ as a map from the plane to the unit circle, compute a truncated Hilbert transform in direction $v$:
$$ H_v f(x)= \int_{-1}^1 f(x-yv(x)) dy/y
$$ Theorem: If $v$ has strictly more than one derivative, then $H_v$ map $L^2$ into itself. The significance of this result is that it
holds in absence of geometric conditions, and under nearly minimal smoothness conditions. The proof is an elaborate variation on a proof of pointwise convergence of Fourier series,
with a novel maximal function of Kakeya type, specifically adapted to the the choice of vector field. (Joint work with Xiaochun Li.)

Presentation (PDF File)

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