On Calder\'on's conjecture

Abstract

This paper is a successor of \\cite{laceyt}. In that paper we considered\nbilinear operators of the form\n H_alpha(f_1,f_2)(x) = p.v. \\int f_1(x-t) f_2(x + alpha t)/t dt,\n which are originally defined for f_1, f_2 in the Schwartz class S(R). The\nnatural question is whether estimates of the form\n H_alpha(f_1,f_2)|_p <= C_{alpha,p_1,p_2} |f_1|_{p_1} |f_2|_{p_2}\n with constants C_{alpha,p_1,p_2} depending only on alpha,p_1,p_2 and p =\np_1p_2/(p_1+p_2) hold. The purpose of the current paper is to extend the range\nof exponents p_1 and p_2 for which the estimate is known. In particular, the\ncase p_1=2, p_2=\\infty is solved to the affirmative. This was originally\nconsidered to be the most natural case and is known as Calder\\'on's conjecture.\n