A symbolic calculus for layer potentials on ¹ curves and ¹ curvilinear polygons

摘要: A symbolic calculus for some algebras of Mellin operators on the finite interval J _ [0, 11 is developed. The are ample enough to include singular integral and analytic double layer potentials their adjoints C1 curves piecewise with corners. Fredholmness index LP(J) completely determined by principal symbol Lp (J), Smbl1/. This note describes pseudodifferential type, or operators, LP(J), 1], which useful potential curvilinear polygons. resulting gives necessary sufficient conditions acting (J) . paper a companion Lewis [L]. first step enlarge described in Lewis-Parenti [LP], J. Elschner [E], [L] multiplication continuous functions C(J). In ? 1 we carefully describe stucture Op 1, 3 then operator class Opla 0 C(J) extend map, Smbll/p. ?2, show that kernel k how calculate symbols. Theorem 2 shows important Hilbert transform Hardy at endpoints corners curves. essential (Theorem 3) use Commutator A. P. Calder6n C. [CCFJR] remainders compact LP(J). pioneering work domains was done E. B. Fabes, M. Jodeit, N. Riviere [FJR]. Cohen Gosselin [CG] treated multiple biharmonic domains. Our proof spirit [FJR] [CG]. Received editors September 21, 1989 and, revised form, March 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 35S15, 45EO5.