HILBERT TRANSFORMS ALONG LIPSCHITZ DIRECTION FIELDS: A LACUNARY MODEL. Issue 2 (9th February 2017) - Guo, Shaoming; Thiele, Christoph
Abstract : We prove bounds for the truncated directional Hilbert transform in L p ( R 2 ) for any 1 < p < ∞ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for p = 2, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some p .