On the bilinear Hilbert transform along two polynomials

Abstract

We prove that the bilinear Hilbert transform along two polynomials B P,Q (f, g)(x) = R f (x−P (t))g(x−Q(t)) dt t is bounded from L p ×L q to L r for a large range of (p, q, r), as long as the polynomials P and Q have distinct leading and trailing degrees. The same boundedness property holds for the corresponding bilinear maximal function M P,Q (f, g)(x) = sup ǫ>0 1 2ǫ ǫ −ǫ |f (x − P (t))g(x − Q(t))|dt.

Two bipolynomial Roth theorems in $\mathbb{R}$

“…This concept was inspired by Gowers' work in [7] and crucial in [14]'s study of bilinear Hilbert transforms along monomials. Such a strategy involving the σ-uniformity was later often used to study variants of the bilinear Hilbert transform, for example, the polynomial case in [15], the general curve case in [8] and the bipolynomial case in [4]. Since we have transformed the bipolynomial to a general curve, our treatment in Section 6 is similar to those in [14,8].…”

Discrete bilinear Radon transforms along arithmetic functions with many common values

“…Lacey–Thiele's breakthrough study of BHT has generated numerous investigations in multilinear operators. One direction of generalizing BHT is to replace the linear terms and with some non‐linear polynomials and , and consider the operator See the articles of Li , Li–Xiao and the first author for some recent results about . Motivated by many works on discrete linear operators (see, for example, Bourgain , Ionescu–Wainger , Krause , Mirek–Trojan , Mirek–Stein–Trojan , Pierce , Zorin–Kranich , etc.…”

“…See the articles of Li [22,23], Li-Xiao [24] and the first author [8] for some recent results about B P,Q . Motivated by many works on discrete linear operators (see, for example, Bourgain [3], Ionescu-Wainger [16], Krause [18], Mirek-Trojan [29], Mirek-Stein-Trojan [27,28], Pierce [30], Zorin-Kranich [35], etc.…”