  We introduce a modified Balitskii-Fadin-Kuraev-Lipatov (BFKL) equation with
the rapidity veto that originates from external kinematic constraint. Though it
is a formally sub-leading power contribution, such kinematic effect becomes
very important especially in the threshold region, where there is no sufficient
phase space for the small $x$ evolution to be fully developed. We also
investigate the phenomenological consequences of the kinematic constraint BFKL
equation in the forward particle production processes in pp and eA collisions.
It is shown that particle yield at large transverse momentum is significantly
suppressed in the threshold region.
