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 \begin{aligned} H_k( f_1,\dots ,f_k )(x) := {\text {p.v.}} \int _\mathbb {R}f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned} for test functions f_1,\dots ,f_k: \mathbb {R}\rightarrow \mathbb {C}. It is conjectured that H_k maps L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R}) whenever 1 < p_1,\dots ,p_k,p < \infty and \frac{1}{p} = \frac{1}{p_1} + \dots + \frac{1}{p_k}. This is proven for k=1,2, but remains open for larger k. In this paper, we consider the truncated operators \begin{aligned} H_{k,r,R}( f_1,\dots ,f_k )(x) := \int _{r \leqslant |t| \leqslant R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t} \end{aligned} for R > r > 0. The above conjecture is equivalent to the uniform boundedness of \Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})} in r, R, whereas the Minkowski and Hölder inequalities give the trivial upper bound of 2 \log \frac{R}{r} for this quantity. By using the arithmetic regularity and counting lemmas of Green and the author, we improve the trivial upper bound on \Vert H_{k,r,R} \Vert _{L^{p_1}(\mathbb {R}) \times \dots \times L^{p_k}(\mathbb {R}) \rightarrow L^p(\mathbb {R})} slightly to o( \log \frac{R}{r} ) in the limit \frac{R}{r} \rightarrow \infty for any admissible choice of k and p_1,\dots ,p_k,p. This establishes some cancellation in the k-linear Hilbert transform H_k, but not enough to establish its boundedness in L^p spaces.