On Calder\'on's conjecture

Abstract

This paper is a successor of \cite{laceyt}. In that paper we considered bilinear operators of the form H_alpha(f_1,f_2)(x) = p.v. \int f_1(x-t) f_2(x + alpha t)/t dt, which are originally defined for f_1, f_2 in the Schwartz class S(R). The natural question is whether estimates of the form H_alpha(f_1,f_2)|_p <= C_{alpha,p_1,p_2} |f_1|_{p_1} |f_2|_{p_2} with constants C_{alpha,p_1,p_2} depending only on alpha,p_1,p_2 and p = p_1p_2/(p_1+p_2) hold. The purpose of the current paper is to extend the range of exponents p_1 and p_2 for which the estimate is known. In particular, the case p_1=2, p_2=\infty is solved to the affirmative. This was originally considered to be the most natural case and is known as Calder\'on's conjecture.