8 TORU TSUJISHITA manifolds M- and M? respectively. The following is well-known. PR0P0SITI0N(1.2.1). There are the following natural isomorphisms: rCVj,)® r(v2) a v(v1 E v2), r (v ) ® & (v2) a (FM®^» (M2) ) ®F r (vx H v2), 1 M, X M2 rcvp ®.^'(v2) a (FM®y(M2)) ®p r(vx a v2). 1 M, X M^ Here FM ®^ f (M2) and FM &$g% (M2) are regarded as FM x M -modules in natural ways. We shall need also the following THEOREM(1.2.2)(cf. [34]). There are natural isomorphisms: ro^} ® r(v2)» e L(r(v2), ro^)), rcv^' ® r(v2)f aL(r(v2), ro^)'). Here L(W1, W2) denotes the space of all the continuous linear maps from W, to W~ endowed with the bounded convergence topology. 1.3. DECOMPOSITION THEOREM. Note that there is a natural inclusion map F^ C^'(M). Let M = R*, the linear space Rn with a fixed linear coordinate x = (x ,...,xn). Then, by the volume element dx * dx . ..dx11, we identify r(KM) = FM, whence j* » (M) - (FM)» (M = R£ , KM = A V ) . The following theorem plays an essential role in Section 8 and will be proved in Section 10. THEOREM (1.3.1). Suppose u ej^»(Rn X Rm) is C°° with respect to x y x, that is, u(FDm) C F^n when it is regarded as a map FDm • (FDn)' . *V K x K y K x Assume N supp u C u v. , j - 1 J

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