Hilbert transform on Vector Fields

speaker: Michael Lacey (Georgia Institute of Technology)

abstract: For a vector field $v$ from the plane to the unit circle we set the Hilbert transform in direction $v$ to be $$ H_v f(x)= \int_{-1}¹ f(x-yv(x)) dyy $$ If $v$ has $1+\epsilon$ derivatives, then $H_v$ is a bounded operator on $L2$. Essential elements of the proof are the proof of Carleson's Theorem of myself and Christoph Thiele, adapted to the current setting of the plane. A novel ingredient is a maximal function which is adapted to choice of vector field. This maximal function admits a favorable bound assuming only that the vector field is Lipschitz. This is joint work with Xiaochun Li.

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