Kernels for the tangential Cauchy-Riemann equations
Authors: Boggess Al 
Published: June 2010

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E colon script upper D Subscript upper S Superscript p comma q Baseline left-parenthesis upper S minus upper C Subscript upper S Baseline right-parenthesis right-arrow script upper E Subscript upper S Superscript p comma q minus 1 Baseline left-parenthesis upper S minus upper C Subscript upper S Baseline right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p comma c o m p Superscript asterisk Baseline left-parenthesis upper S minus upper C Subscript upper S Baseline right-parenthesis right-arrow upper L Subscript p comma l o c Superscript asterisk Baseline left-parenthesis upper S minus upper C Subscript upper S Baseline right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f equals ModifyingAbove partial-differential With bar Subscript upper S Baseline left-brace upper E left-parenthesis f right-parenthesis right-brace plus upper E left-parenthesis ModifyingAbove partial-differential With bar Subscript upper S Baseline f right-parenthesis">

f

=

∂
¯

S

{
E
(
f
)
}

+

E
(

∂
¯

S

f
)

f\, = \,{\bar \partial _S}\{ E(f)\} \, + \,E({\bar \partial _S}f)

.