Subelliptic estimates for complexes.
New results are announced linking properties of the symbol module and characteristic variety of a differential complex with test estimates near the characteristic variety of the type considered by Hörmander ((1/2)-estimate). The first result is the invariance of the test estimates under pseudo-differential change of coordinates, and this leads to the introduction of a normal form for the complex in the neighborhood of a Cohen-MacCauley point of the symbol module. If the characteristic variety V is a manifold near the Cohen-MacCauley point (x(0),zeta(0)) with parametrizing functions p(1),...,p(q), where q is the codimension of the characteristic variety in the complexified contangent bundle, the matrix [Formula: see text] of Poisson brackets defines invariantly a Hermitian form Q on the normal space to V at (x(0),zeta(0)) when the dp(zeta)(x(0),zeta(0)) are used as basis, and the test estimates are satisfied at the ith stage of the complex if sig. Q (signature of Q) is >/= n - i + 1 (n the dimension of the base manifold) or rank Q - sig. Q >/= i + 1. Finally, conditions are given in order that, on a manifold with smooth boundary, the associated boundary complexes satisfy the (1/2)-estimate.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY, HARVARD UNIVERSITY.
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Guillemin
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Sternberg