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arXiv · 1902.10577

Modulation invariant operators

Pavel Zorin-Kranich
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Abstract

The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, in a dyadic model we give a partial result in the trilinear, 2-dimensional case when one of the functions depends only on one variable. Thirdly, we estimate the multidimensional version of the polynomial Carleson operator, whose boundedness would also be a consequence of the conejcture. The final pair of results concerns directional square functions. One of them concerns interaction of Lipschitz change of variable on the line with Littlewood--Paley decomposition and is used to extend the conditional result of Lacey and Li for Hilbert transform along $C^{1+\epsilon}$ vector fields to Lipschitz vector fields. The other concerns directional averaging operators and gives an alternative approach to a single scale maximal function with averages in N directions due to Katz away from the endpoint.

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Pavel Zorin-Kranich. 2019-02-27. Modulation invariant operators. https://arxiv.org/abs/1902.10577

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