Cancellation for the multilinear Hilbert transform

Autores: Terence Tao
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 67, Fasc. 2, 2016, págs. 191-206
Idioma: inglés
DOI: 10.1007/s13348-015-0162-y
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Resumen
- For any natural number k, consider the k-linear Hilbert transform ??(?1,…,??)(?):=p.v.∫ℝ?1(?+?)…??(?+??) ??? for test functions ?1,…,??:ℝ→ℂ . It is conjectured that ?? maps ??1(ℝ)×⋯×???(ℝ)→??(ℝ) whenever 1?>0 . The above conjecture is equivalent to the uniform boundedness of ‖??,?,?‖??1(ℝ)×⋯×???(ℝ)→??(ℝ) in r, R, whereas the Minkowski and Hölder inequalities give the trivial upper bound of 2log?? for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on ‖??,?,?‖??1(ℝ)×⋯×???(ℝ)→??(ℝ) slightly to ?(log??) in the limit ??→∞ for any admissible choice of k and ?1,…,??,? . This establishes some cancellation in the k-linear Hilbert transform ?? , but not enough to establish its boundedness in ?? spaces.