Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

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 point p ∈ bD 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.