  In this paper we prove a sharp trilinear inequality which is motivated by a
program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint
restriction inequality on the circle. Our method uses intricate estimates for
integrals of sixfold products of Bessel functions developed in a companion
paper. We also establish that constants are local extremizers of the
Tomas-Stein adjoint restriction inequality as well as of another inequality
appearing in the program.
