In troduc tiori
Let ( M , g) be a pseudo-RIEMANNian C" manifold of dimension n ( n z 4 ) and D(g) a polynomial conformally invariant linear differential operator, i.e. a linear operator with the following properties:
(i) D(g) acts on a C" tensor field 5 of any type which is defined in an open set
U C M .
(ii) Let the coefficients of D(g) be polynomials in the covariant and contravariant coordinates and their partial derivatives of the metric tensor g.
(iii) Under a conformal change of the metric 0 = e'@g, (@€ C"( U ) ) the operator D(g) transforms according to D(g) [e%@,] = + W O P D(g) [ u ] for ZLE% and some w , w , , € R .
The polynomial conformally invariant differential operators have a great variety of applications, e.g. in the representation theory of the conformal group C ( X , g) [3, 5, 8, 14, 211 g), in the "conformal extension" of the heat equation [6, 71, in connection with the investigations of the YAMABE problem [6, 26, 11, on HUYGENS' principle [11, 17, 5 , 23, 241, and of zero-mass field equations 1119, 231. It is an important problem to give a survey of all conformally invariant differential operators or, with less pretention, to give a method for constructing special classes of such operators. The study of these operators was inspired by the following: ( 1 ) the investigations on such operators by B. ~R S T E D 116, 17, 71, T. P. BRANSON [4. 5 , 6, 71 and S. PANEITZ [18], (2) the contributions to the theory of conformally invariant tensors and the construction of such tensors by P. GUEN-THER and the author 112, 13, 2.21, (3) the result found by H. P. JAKOBSON and M. VERGNE [ 141 that in the special case of four-dimensional MINKOWSKI space the powers (V"V,)" are invariant under the conformal group, and (4) the sequences of conformal invariants for gauge fields, derived by R. SCHIMMING [2O]. Using the basic ideas a i d results on conformally invariant tensors [ 12. 131, in particular, 1 ) The conformally invariant differential opmxtors act. i ~s int.crtn ining operators for cert.ain tensor representations of C ( M , g ) [ci, 61. Remarks. (i) The CLDO (3.11) is that one introduced by T.BRANSON in (ii) I n the case m0 = 0 we obtain the conformally invariant MAXWELL operator 1982 [4]. c c 1 -D tz =6d=6d. 4 ?,---I 2 1 is for n > 4 a CLDO on Ap with co= -2, w o = z ( 4 f 2 p -n ) .
Proof. Because of (3.10), (3.12), (3.13) and 2 p -n = 4 (wo-1) it holds