Title: Pseudodifferential Calculus at the Critical Line II: Functional Calculus and Coordinate Invariance

Abstract: At the critical indices (\rho=\delta=1/2), the usual symbolic expansions no longer yield terms of successively lower order. How can we establish a pseudodifferential calculus without relying on such expansions? Continuing the first talk, I will address this question through two related proofs. First, I will explain how Taylor’s wave-group approach and integration by parts adapted to the critical scale yield a local pseudodifferential description of (p(A)), where (p\in S^m_{1/2}(\mathbb R)) and (A) is a nonnegative elliptic self-adjoint operator of order one in a suitable Euclidean setting. Second, I will discuss the reduction of amplitudes to symbols and use it to establish invariance of the class (\Psi^m_{1/2,1/2}) under smooth changes of coordinates, giving an intrinsic local definition on smooth manifolds. The emphasis will be on the common analytical mechanism: critical rescaling and direct integration-by-parts estimates for oscillatory integrals. A brief review of the necessary definitions and the motivation from the first talk will be included.