Singular Integral Operators on C ¹ Manifolds and C ¹ Curvilinear Polygons

Summary

The Multilinear Commutator Theorem of Alberto Calderón, Calixto Calderón, Eugene Fabes, Max Jodeit, and Nestor Rivière (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).