Kernels for the tangential Cauchy-Riemann equations

Author:
Al Boggess

Journal:
Trans. Amer. Math. Soc. **262** (1980), 1-49

MSC:
Primary 32F20; Secondary 35N15

DOI:
https://doi.org/10.1090/S0002-9947-1980-0583846-0

MathSciNet review:
583846

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Abstract: On certain codimension one and codimension two submanifolds in , we can solve the tangential Cauchy-Riemann equations with an explicit integral formula for the solution.

Let , where *D* is a strictly pseudoconvex domain in . Let be defined by , where *h* is holomorphic near *D*. Points on the boundary of , , where the tangent space of becomes complex linear, are called characteristic points.

Theorem 1. *Suppose* *is admissible (in particular if* *has two characteristic points). Suppose* , , *is smooth on* *and satisfies* *on* ; *then there exists* *which is smooth on* *except possibly at the characteristic points on* *and which solves the equation* on .

Theorem 2. *Suppose* , , *is smooth on* ; *vanishes near each characteristic point; and* *on* . *Then there exists* *satisfying* *on* .

Theorem 3. *Suppose* , , *is smooth with compact support in* , *and* . *Then there exists* *with compact support in* *and which solves* .

In all three theorems we have an explicit integral formula for the solution.

Now suppose . Let be the set of characteristic points on *S*. We construct an explicit operator with the following properties.

Theorem 4. *The operator E maps* *and if* , , *then* .

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0583846-0

Article copyright:
© Copyright 1980
American Mathematical Society