TY - JOUR

T1 - Kernels for the tangential cauchy-riemann equations

AU - Boggess, A.

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1980/9

Y1 - 1980/9

N2 - On certain codimension one and codimension two submanifolds in Cn, we can solve the tangential Cauchy-Riemann equations (FORMULA PRESENTED) with an explicit integral formula for the solution. Let (FORMULA PRESENTED), where D is a strictly pseudoconvex domain in (FORMULA PRESENTED) M be defined by (FORMULA PRESENTED), where h is holomorphic near D. Points on the boundary of ω, ¶ω, where the tangent space of ¶ω becomes complex linear, are called characteristic points. Theorem 1. Suppose ¶ω is admissible (in particular if ¶ω has two characteristic points). Suppose (FORMULA PRESENTED), is smooth on ω and satisfies (FORMULA PRESENTED) on ω then there exists (FORMULA PRESENTED) which is smooth on ω except possibly at the characteristic points on ¶ω and which solves the equation (FORMULA PRESENTED) on ω. Theorem 2. Suppose (FORMULA PRESENTED), is smooth on ω; vanishes near each characteristic point, and (FORMULA PRESENTED) then there exists (FORMULA PRESENTED) satisfying (FORMULA PRESENTED). Theorem 3. Suppose (FORMULA PRESENTED), is smooth with compact support in and (FORMULA PRESENTED). Then there exists (FORMULA PRESENTED) compact support in ω and which solves (FORMULA PRESENTED). In all three theorems we have an explicit integral formula for the solution. Now suppose S = ¶ω. Let Cs be the set of characteristic points on S. We construct an explicit operator (FORMULA PRESENTED) with the following properties. Theorem 4. The operator E maps (FORMULA PRESENTED).

AB - On certain codimension one and codimension two submanifolds in Cn, we can solve the tangential Cauchy-Riemann equations (FORMULA PRESENTED) with an explicit integral formula for the solution. Let (FORMULA PRESENTED), where D is a strictly pseudoconvex domain in (FORMULA PRESENTED) M be defined by (FORMULA PRESENTED), where h is holomorphic near D. Points on the boundary of ω, ¶ω, where the tangent space of ¶ω becomes complex linear, are called characteristic points. Theorem 1. Suppose ¶ω is admissible (in particular if ¶ω has two characteristic points). Suppose (FORMULA PRESENTED), is smooth on ω and satisfies (FORMULA PRESENTED) on ω then there exists (FORMULA PRESENTED) which is smooth on ω except possibly at the characteristic points on ¶ω and which solves the equation (FORMULA PRESENTED) on ω. Theorem 2. Suppose (FORMULA PRESENTED), is smooth on ω; vanishes near each characteristic point, and (FORMULA PRESENTED) then there exists (FORMULA PRESENTED) satisfying (FORMULA PRESENTED). Theorem 3. Suppose (FORMULA PRESENTED), is smooth with compact support in and (FORMULA PRESENTED). Then there exists (FORMULA PRESENTED) compact support in ω and which solves (FORMULA PRESENTED). In all three theorems we have an explicit integral formula for the solution. Now suppose S = ¶ω. Let Cs be the set of characteristic points on S. We construct an explicit operator (FORMULA PRESENTED) with the following properties. Theorem 4. The operator E maps (FORMULA PRESENTED).

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U2 - 10.1090/S0002-9947-1980-0583846-0

DO - 10.1090/S0002-9947-1980-0583846-0

M3 - Article

AN - SCOPUS:84967735787

VL - 262

SP - 1

EP - 49

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -