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In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group Hn, whose principal parts are “positive combinations of generalized and degenerate generalized subLaplacians”, and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete way. Previously known methods to study such non-homogeneous operators, as in [9] or [6], do not apply to these operators, and it is the main purpose of this article to introduce a new method, which should be applicable also in much wider settings. 1. Basic definitions and main results. Let ω denote the symplectic form on R2n given by ω(z, z′) := tz′Jz, J = Jn := ( 0 In −In 0 ) . The Heisenberg group Hn is R2n × R, endowed with the group law (z, u)(z′, u) = ( z + z′, u+ u′ − 2ω(z, z′) ) . (1.1) The left-invariant vector fields Xj := ∂ ∂xj − 1 2 yj ∂ ∂u , Yj := ∂ ∂yj + 1 2 xj ∂ ∂u , j = 1, . . . , n, and U := ∂/∂u form a natural basis for the Lie algebra hn of Hn. The only non-trivial commutation relations among these vector fields are [Xj, Yj] = U , j = 1, . . . , n. Denote by sp(n,C) the symplectic Lie algebra, consisting of all complex 2n× 2n-matrices S satisfying tSJ + JS = 0. 2000 Mathematics Subject Classification: 35A05, 43A80. Work of the second author is supported by NNSF of China and NSF of Tianjin University.

A class of solvable non-homogeneous differential operators on the Heisenberg group