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Extra info for Algebraic geometry and its applications
Homogeneous symplectic and contact structures 51 where the form is dctlned by formula (6). It is evident that f is an R1-invuriant diffeomorphism and that diagram (15) is commutative for S = S2, = ir2. Let w be symplectic forms on S and S,. respectively, and ww be the form on S and defined by equality (14). Then co=dwW; w1 Evidently, the relation = = is valid. Hence f(w) = = = d((Xo)1jw1) = = WI, f is a symplectic diffeomorphism. To prove the uniqueness of the diffeomorphismf, it suffices to show that if S1 = S2 = 5, r1 = 3T2 = r, and f is a symplectic diffeomorphism such that diagram (15) is commutative, then f is an identity diffeomorphism.
Thus, regf E (22) in other words, the following assertion is valid: The operator (23) —+ is an isomorphism fork> —n — 1 if n + a is odd, and for every k n + a is even. Its inverse is given by the operator reg. if k —n — I and n + a is odd, the operator (23) possesses a kernel and cokernel of equal finite dimension. The kernel consists of linear combinations of the derivatives of the Dirac ö-function of order —(n + k + I). The cokernel is defined by the orthogonality conditions (xa, f) = 0, a = —n — k — 1.
J Thus we have proved the following theorem. (38) 40 1. Homogeneous functions. Fourier transformation, and contact structures Theorem I. Let the continuous mapping F (39) satisfy the condition (2) and the commutation conditions of the type (33). Then this mapping is given bvfor,nula (38) or one of the last three formulas in (32), depending on k, n. and a. 9 The inversion formulas and the Parseval identity Just as for the usual Fourier transformation, the inverse transformation may be obtained by substituting —i instead of i in all the formulas.