Kernels for the tangential Cauchy-Riemann equations

Abstract

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