2015/05/24 by Terence Tao, Tao, Terence
Mathematics · #11B30 #42B20 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #math.CA #math.CO #msc:11B30 #msc:42B20
paper · pdf · doi:10.48550/arxiv.1505.06479
17 pages, no figures. An error pointed out to the author by Pavel Zorin-Kranich has been corrected
openalex publication_date 2015/05/24 · arxiv created 2015/05/29 · arxiv updated 2015/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any natural number k, consider the k-linear Hilbert transform Hk( f1,…,fk )(x) := p.v. ∫\bf R f1(x+t) … fk(x+kt) (dt)/(t) for test functions f1,…,fk: \bf R → \bf C. It is conjectured that Hk maps Lp1(\bf R) × … × Lpk(\bf R) → Lp(\bf R) whenever 1 < p1,…,pk,p < ∞ and (1)/(p) = (1)/(p1) + … + (1)/(pk). This is proven for k=1,2, but remains open for larger k. In this paper, we consider the truncated operators Hk,r,R( f1,…,fk )(x) := ∫r ≤ |t| ≤ R f1(x+t) … fk(x+kt) (dt)/(t) for R > r > 0. The above conjecture is equivalent to the uniform boundedness of ‖ Hk,r,R ‖_Lp1(\bf R) × … × Lpk(\bf R) → Lp(\bf R) in r,R, whereas the Minkowski and Hölder inequalities give the trivial upper bound of 2 log (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 ‖ Hk,r,R ‖_Lp1(\bf R) × … × Lpk(\bf R) → Lp(\bf R) slightly to o( log (R)/(r) ) in the limit (R)/(r) → ∞ for any admissible choice of k and p1,…,pk,p. This establishes some cancellation in the k-linear Hilbert transform Hk, but not enough to establish its boundedness in Lp spaces.