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We run an iteration argument due to Pramanik and Seeger, to provide a proof of sharp decoupling inequalities for conical surfaces and for k-cones. These are extensions of results of Łaba and Pramanik to sharp exponents.
We prove bounds for the truncated directional Hilbert transform in
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
, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some
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