In this paper we completely describe the automorphism group of the Hartogs type domains over bounded classical symmetric domains by using the ball characterization theorem about noncompact automorphism groups. As an application, we give plenty of examples of non-homogeneous domains in ℂn, for which the real dimension of the automorphism group is less than n2 - 1.

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Motivated by the Kohn-Nirenberg domain, J.E. Fornaess considered the germ of a domain near the origin in C 2 such that Ω t = (z, w) ∈ C 2 : Re w + |zw| 2 + |z| 6 + t|z| 2 Re z 4 < 0 to study the holomorphic peak function that is smooth up to the boundary (Fornaess, 1977 [6]). J.E. Fornaess proved that for 1 < t < 9/5 the domain Ω t does not admit a holomorphic function on Ω t that is C 1 up to the boundary and that peaks at the origin. We define Π(z, w) = (e i π 2 z, w). In this paper, we prove that for 1 < t < 9/5, the automorphism group of Ω t is equal to the set {Π k : k = 1, 2, 3, 4}.

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