Remarks on lower bounds for pseudo-differential operators

We study lower bounds for pseudo-differential operators with multiple characteristics. The principal symbol is assumed positive, vanishing exactly to the order k greater than or equal to 2 on a smooth manifold Sigma. Under an additional positivity assumption on the Jth Taylor polynomial of the sub-principal symbol at Sigma, 0 less than or equal to J less than or equal to k/2 - 1, using the Fefferman-Phong inequality we get a lower bound with gain of k/(k - J - 1) derivatives. (C) 2004 Elsevier SAS. All rights reserved.