  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.
