Kernels for the Tangential Cauchy-Riemann Equations

Abstract

On certain codimension one and codimension two submanifolds in ${{\textbf {C}}^n}$, we can solve the tangential Cauchy-Riemann equations ${\bar \partial _b}u = f$ with an explicit integral formula for the solution. Let $M = \partial D$, where D is a strictly pseudoconvex domain in ${{\textbf {C}}^n}$. Let $\omega \subset \subset M$ be defined by $\omega = \{ z \in M; \operatorname {Re} h(z) < 0\}$, where h is holomorphic near D. Points on the boundary of $\omega$, $\partial \omega$, where the tangent space of $\partial \omega$ becomes complex linear, are called characteristic points. Theorem 1. Suppose $\partial \omega$ is admissible (in particular if $\partial \omega$ has two characteristic points). Suppose $f \in {\mathcal {E}}_M^{p,q}(\bar \omega )$, $1 \leqslant q \leqslant n - 3$, is smooth on $\omega$ and satisfies ${\bar \partial _M}f = 0$ on $\omega$; then there exists $u \in {\mathcal {E}}_M^{p,q - 1}(\omega )$ which is smooth on $\omega$ except possibly at the characteristic points on $\partial \omega$ and which solves the equation ${\bar \partial _M}u = f$ on $\omega$. Theorem 2. Suppose $f \in {\mathcal {E}}_M^{p,q}(\omega )$, $2 \leqslant q \leqslant n - 3$, is smooth on $\omega$; vanishes near each characteristic point; and ${\bar \partial _M}f = 0$ on $\omega$. Then there exists $u \in {\mathcal {E}}_M^{p,q - 1}(\omega )$ satisfying ${\bar \partial _M}u = f$ on $\omega$. Theorem 3. Suppose $f \in {\mathcal {D}}_M^{p,q}(\omega )$, $2 \leqslant q \leqslant n - 3$, is smooth with compact support in $\omega$, and ${\bar \partial _M}f = 0$. Then there exists $u \in {\mathcal {D}}_M^{p,q - 1}(\omega )$ with compact support in $\omega$ and which solves ${\bar \partial _M}u = f$. In all three theorems we have an explicit integral formula for the solution. Now suppose $S = \partial \omega$. Let ${C_s}$ be the set of characteristic points on S. We construct an explicit operator $E: {\mathcal {D}}_S^{p,q}(S - {C_S}) \to {\mathcal {E}}_S^{p,q - 1}(S - {C_S})$ with the following properties. Theorem 4. The operator E maps $L_{p,\operatorname {comp} }^{\ast }(S - {C_S}) \to L_{p,\operatorname {loc} }^{\ast }(S - {C_S})$ and if $f \in {\mathcal {D}}_S^{p,q}(S - {C_S})$, $1 \leqslant q \leqslant n - 3$, then $f = {\bar \partial _S}\{ E(f)\} + E({\bar \partial _S}f)$.