A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators

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 set K subset of X, an open set of R(n), and P = P* is an element of Psi(phg)(4)(X) with p(x, xi) similar to q(2)(2) + p(3) + p(2) + ..., q(2) being transversally elliptic with respect to the characteristic manifold Sigma = q(2)(-1)(0).