  Let D be a J-pseudoconvex region in a smooth almost complex manifold (M,J) of
real dimension four. We construct a local peak J-plurisubharmonic function at
every boundary point p of finite D'Angelo type. As applications we give local
estimates of the Kobayashi pseudometric, implying the local Kobayashi
hyperbolicity of D at p. In case the point p is of D'Angelo type less than or
equal to four, or the approach is nontangential, we provide sharp estimates of
the Kobayashi pseudometric.
