By Glenn O. E.

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In Russian). [17] M. Langenbruch, Hermite functions and weighted spaces of generalized functions, Manuscripta Math. 119 (2006), 269–285. [18] W. Magnus, F. G. Tricomi, Higher Transcendental Functions II, Bateman Project, California Institute of Technology, McGraw–Hill, 1953. S. Mitjagin, Nuclearity and other properties of spaces of type S, Amer. Math. Soc. Transl. Ser. 2 93 (1970), 45–59. Degenerate Elliptic Operators 31 [20] S. Pilipovi´c, Generalization of Zemanian spaces of generalized functions which elements have series expansion, SIAM J.

Here B(Rn ) is the set of smooth functions in Rn whose all partial derivatives are bounded. We deﬁne a pseudo-diﬀerential operator P = p(x, D : A) acting on A∗ (M ) of a symbol σ(P ) = p(x, ξ : A) = I,J pI,J (x, ξ)a∗I aJ ∈ K m as follows. pI,J (x, D)ϕK a∗I aJ (ω K ). p(x, D : A)(ϕK ω K ) = I,J 50 C. 1 the symbol of Δ is of the form σ(Δ) = r2 + r1 , where n r2 = − (αj I − Gj )2 + R. j=1 j It is clear that rj belongs to K for j = 1, 2. If we review the results in [11], we see the following facts.

N) of order m with coeﬃcients in B(Rn )}. Here B(Rn ) is the set of smooth functions in Rn whose all partial derivatives are bounded. We deﬁne a pseudo-diﬀerential operator P = p(x, D : A) acting on A∗ (M ) of a symbol σ(P ) = p(x, ξ : A) = I,J pI,J (x, ξ)a∗I aJ ∈ K m as follows. pI,J (x, D)ϕK a∗I aJ (ω K ). p(x, D : A)(ϕK ω K ) = I,J 50 C. 1 the symbol of Δ is of the form σ(Δ) = r2 + r1 , where n r2 = − (αj I − Gj )2 + R. j=1 j It is clear that rj belongs to K for j = 1, 2. If we review the results in [11], we see the following facts.