Jeff E. Lewis

Propagation of Singularities for Fuchsian Operators

Preliminaries and review of results of N. Hanges.- General fuchsian systems.- Applications to Fuchsian hyperbolic P.D.E..- Operators with multiple non-involutive characteristics.

Singular Integral Operators on C1 Manifolds and C1 Curvilinear Polygons

The Multilinear Commutator Theorem of Alberto Calderon, Calixto Calderon, Eugene Fabes, Max Jodeit, and Nestor Riviere (Indiana Univ. Math. J. 49(1), 1–5, 2000; Bull. Am. Math. Soc. 84(2):287–290, 1978) can be used to show that an Analytic Double Layer Kernel restricted to the boundary of a C 1 domain fits into the symbolic calculus for singular integral operators (S.I.O.) with C 0 coefficients on C 1 manifolds and C 1 Curvilinear Polygons. In particular a system of S.I.O. is Fredholm on L p if and only the symbol is a nonsingular homomorphism. For C 1 curvilinear polygons, the symbol is constructed using the Mellin transform and can be extended to handle power weights near the vertices. The talk is a review of joint work with Renata Selvaggi and Irene Sisto (Trans. Am. Math. Soc. 340(1), 293–308 (1993), MR MR1124170 (94a:58194)) and joint work with Lewis and Cesare Parenti (Commun. Partial Differ. Equ 8(5), 477–544, 1983, MR MR695401, 86f:35185), Lewis (Proc. Am. Math. Soc. 112(2), 419–427, 1991, MR MR1043413, 91i:47071).