Discrete bilinear Radon transforms along arithmetic functions with many common values. Issue 1 (1st December 2017)

• Record Type: Journal Article
• Authors: Dong, Dong; Meng, Xianchang
• Is Part Of: Bulletin of the London Mathematical Society. Volume 50:Issue 1(2018)
• Page Start: 132
• Page End: 142
• Publication Date: 2017-12-01
• Subjects: 42B20 -- 11N64 -- 11N05 (primary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12127
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: We prove that for a large class of functions P and Q, there exists d ∈ ( 0, 1 ) such that the discrete bilinear Radon transform B P, Q dis ( f, g ) ( n ) = ∑ m ∈ Z ∖ { 0 } f ( n − P ( m ) ) g ( n − Q ( m ) ) 1 m is bounded from l 2 × l 2 into l 1 + ε for any ε ∈ ( d, 1 ) . In particular, the boundedness holds for any ε ∈ ( 0, 1 ) when P (or Q ) is the Euler totient function ϕ ( | m | ) or the prime counting function π ( | m | ) .