Improved estimates for bilinear rough singular integrals

• Published in: Mathematische annalen 2023-08, Vol.386 (3-4), p.1951-1978
• Main Authors: He, Danqing, Park, Bae Jun
• Format: Article
• Language: English
• Subjects: Estimates; Mathematics; Mathematics and Statistics; Operators (mathematics)
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-022-02444-2

We study bilinear rough singular integral operators L Ω associated with a function Ω on the sphere S 2 n - 1 . In the recent work of Grafakos et al. (Math Ann 376:431–455, 2020), they showed that L Ω is bounded from L 2 × L 2 to L 1 , provided that Ω ∈ L q ( S 2 n - 1 ) for 4 / 3 < q ≤ ∞ with mean value zero. In this paper, we provide a generalization of their result. We actually prove L p 1 × L p 2 → L p estimates for L Ω under the assumption Ω ∈ L q ( S 2 n - 1 ) for max ( 4 3 , p 2 p - 1 ) < q ≤ ∞ where 1 < p 1 , p 2 ≤ ∞ and 1 / 2 < p < ∞ with 1 / p = 1 / p 1 + 1 / p 2 . Our result improves that of Grafakos et al. (Adv Math 326:54–78, 2018), in which the more restrictive condition Ω ∈ L ∞ ( S 2 n - 1 ) is required for the L p 1 × L p 2 → L p boundedness.