2021/04/29 by Danqing He, Bae Jun Park, He, Danqing +1 · 5 citations
Mathematics · #42B20 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #math.CA #msc:42B20
paper · pdf · doi:10.48550/arxiv.2104.14137
Minor revision. To appear in Math. Ann
openalex publication_date 2021/04/29 · arxiv created 2022/07/13 · arxiv updated 2022/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study bilinear rough singular integral operators LΩ associated with a function Ω on the sphere \mathbbS2n-1. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that LΩ is bounded from L2× L2 to L1, provided that Ω∈ Lq(\mathbbS2n-1) for 4/3<q≤ ∞ with mean value zero. In this paper, we provide a generalization of their result. We actually prove Lp1× Lp2→ Lp estimates for LΩ under the assumption Ω∈ Lq(\mathbbS2n-1) for ~max( (4)/(3) , (p)/(2p-1) )<q≤ ∞ where 1<p1,p2≤∞ and 1/2<p<∞ with 1/p=1/p1+1/p2 . Our result improves that of Grafakos, He, and Honzík (Adv. Math. 326: 54-78, 2018), in which the more restrictive condition Ω∈ L∞(\mathbbS2n-1) is required for the Lp1× Lp2→ Lp boundedness.