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

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