NECESSARY AND SUFFICIENT CONDITIONS FOR THE ALMOST-POSITIVITY OF CERTAIN 4TH-ORDER PSEUDODIFFERENTIAL-OPERATORS - A GENERALIZATION

We give necessary and sufficient conditions for the lower bound (Pu,u) greater than or equal to-C(K)parallel to u parallel to(1/2)(2), C-K>0, For All u is an element of C-0(infinity)(K) to hold for any compact s et K subset of X, an open set of R-n, for certain pseudodifferential o perators P = P is an element of Psi(phg)(4)(X) with p (chi, zeta) sim ilar to Sigma(i)(m)(q(2)(1))(2) + p(3) + p(2) +..., Sigma(i) q(2)(i) b eing transversally elliptic with respect to the characteristic manifol d Sigma = boolean AND(i)(q(2)(1))(-1)(0).