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p(x, ξ) ∼ pμ(x, ξ) + pμ−1(x, ξ) + . . . , where pμ−j(x, ξ) is positively homogeneous of degree μ− j with respect to ξ. We assume that the characteristic set Σ = p−1 μ (0) of P is a symplectic real analytic submanifold of T ∗(Ω)\0 of codimension 2d and that pμ vanishes exactly at the order m on Σ. As in Grusin [4], Sjostrand [11] and Metivier [8], we also assume that pμ−j vanishes at the order m− 2j on Σ for j ≤ m/2. C∞ and analytic hypoellipticity of this class of operators has been extensively studied by many mathematicians (see e.g., [1], [2], [4], [8], [9], [11], [13] and others). Among them Metivier [8] has proved analytic hypoellipticity of P by constructing a left parametrix when P is subelliptic with loss of m/2 derivatives. In this note, we study hypoellipticity and local solvability of P at a point where the above subellipticity condition is not satisfied. We shall then construct a system of analytic pseudo-differential operators on RN−d to which we can reduce the study of analytic hypoellipticity and local solvability of P . Typical examples of the operators are

Analytic hypoellipticity and local solvability for a class of pseudo-differential operators with symplectic characteristics