Discrete bilinear Radon transforms along arithmetic functions with many common values

TL;DR

This paper establishes boundedness results for a class of discrete bilinear Radon transforms along certain arithmetic functions, including Euler's totient and prime counting functions, extending understanding of their harmonic analysis properties.

Contribution

It proves boundedness of the discrete bilinear Radon transform for a broad class of functions, notably including the Euler totient and prime counting functions, which was previously unknown.

Findings

Boundedness from l^2 x l^2 to l^{1+ε} for ε in (d,1) for a large class of functions.

Special cases where P or Q is the Euler totient or prime counting function are bounded for all ε in (0,1).

Extension of bilinear Radon transform analysis to arithmetic functions with many common values.

Abstract

We prove that for a large class of functions $P$ and $Q$, there exists $d\in (0,1)$ such that the discrete bilinear Radon transform $$B^{\rm dis}_{P,Q}(f,g)(n)=\sum_{m\in\mathbb{Z}\setminus\{0\}} f(n-P(m))g(n-Q(m))\frac{1}{m}$$ is bounded from $l^2\times l^2$ into $l^{1+\epsilon}$ for any $\epsilon\in (d,1)$. In particular, the boundedness holds for any $\epsilon\in (0,1)$ when $P$ (or $Q$) is the Euler totient function $\phi(|m|)$ or the prime counting function $\pi(|m|)$.