Cauchy-Pompeiu type formulas for d-bar on affine algebraic Riemann surfaces and some applications

0.2 MB·English

by  Gennadi Henkin

Cauchy-Pompeiu type formulas for ∂¯ on
affine algebraic Riemann surfaces and some applications
G.M.Henkin
Dedicated to Oleg Viro on the occasion of his 60th birthday
0 Abstract
1
0 We present explicit solution formulas f = Rˆϕ and u = Rλf for the equations ∂¯f = ϕ
2 and (∂ + λdz )u = f f on an affine algebraic curve V C2. Here f denotes
1 − Hλ ⊂ Hλ
n ˜ 1,p˜
the projection of f W (V) to the subspace of pseudoholomorphic (1,0)-forms on V:
a ∈ 1,0
J ∂¯ f = λ¯dz¯ f. These formulas can be interpreted as explicit versions and preci-
Hλ 1 ∧ Hλ
0 sions of the Hodge–Riemann decomposition on Riemann surfaces. The main application
1
¯
consists in the construction of the Faddeev–Green function for ∂(∂ + λdz ) on V as the
1
] kernel of the operator R Rˆ. This Faddeev–Green function is the main tool for the so-
V λ ◦
lution of the inverse conductivity problem on bordered Riemann surfaces X V, that is,
C ⊂
for the reconstruction of the conductivity function σ in the equation d(σdcU) = 0 from
.
h the Dirichlet-to-Neumann mapping U σdcU . The case V = C was treated by
at bX 7→ bX
R.Novikov [N1]. In 4 we give a cor(cid:12)rection to th(cid:12)e paper [HM], in which the case of a
m § (cid:12) (cid:12)
general algebraic curve V was first considered.
[
KeywordsRiemannsurface. Inverseconductivityproblem. ∂¯-method. Homotopyformulas.
2
v Mathematical Subject Classification (2000) 32S65. 32V20. 35R30. 58J32.
1
6
Introduction
7
3
This paper is motivated by a problem from two-dimensional Electrical Impedance
.
4
Tomography, namely the question of how to reconstruct the conductivity function σ on a
0
8 bordered Riemann surface X from the knowledge of the Dirichlet-to-Neumann mapping
0 u σdcU for solutions U of the Dirichlet problem:
: bX → bX
v (cid:12) (cid:12)
i (cid:12) d(cid:12)(σdcU) = 0, U = u, where d = ∂ +∂¯, dc = i(∂¯ ∂).
X X bX −
(cid:12) (cid:12)
ar For the case X = Ω (cid:12) R2 C (z(cid:12) = x1 +ix2) the exact reconstruction scheme was given
⊂ ≃
firstly by R.Novikov [N1] under some restriction on the conductivity function σ. This
restriction was eliminated later by A.Nachman [Na].
The scheme consists in the following.
Let σ(x) > 0 for x Ω¯ and σ C(2)(Ω¯). Putσ(x) = 1 for x R2 Ω¯. The substitution
∈ ∈ ∈ \
ψ = √σU transforms the equation d(σdcU) = 0 into the equation ddcψ = ddc√σψ on R2.
√σ
FromL.Faddeev’s[F1]result(withadditionalarguments[BC2]and[Na])itfollowsthat, for
each λ C, there exists a unique solution ψ(z,λ) of the above equation, with asymptotics
∈
ψ(z,λ) e λz d=ef µ(z,λ) = 1+o(1), z .
−
· → ∞
Such a solution can be found from the integral equation
i µ(ξ,λ)ddc√σ
µ(z,λ) = 1+ g(z ξ,λ) ,
2 Z − √σ
ξ Ω
∈
1
where the function
1 ei(wz¯+w¯z)dw dw¯
g(z,λ) = − ∧ , z C, λ C,
2i(2π)2 Z w(w¯ iλ) ∈ ∈
C −
w
∈
¯
is called the Faddeev–Green function for the operator µ ∂(∂ +λdz)µ.
7→
From work of R.Novikov [N1] it follows that the function ψ can be found through
bΩ
the Dirichlet-to-Neumann mapping by the integral equation (cid:12)
(cid:12)
ψ(z,λ) = eλz + eλ(z ξ)g(z ξ,λ)(Φˆψ(ξ,λ) Φˆ ψ(ξ,λ)),
bΩ Z − − − 0
(cid:12)
(cid:12) ξ bΩ
∈
where Φˆψ = ∂¯ψ , Φˆ ψ = ∂¯ψ , ψ = ψ, ∂¯∂ψ = 0.
bΩ 0 0 bΩ 0 bΩ 0 Ω
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
By results of R.Beals, R.Coifmann [BC1], P.Grinevich, S.Novikov [GN] and R.Novikov
¯
[N2] it then follows that ψ(z,λ) satisfies a ∂-equation of Bers–Vekua type with respect to
λ C:
∈ ∂ψ
¯
= b(λ)ψ,
¯
∂λ
where λ b(λ) L2+ε(C) L2 ε(C), and ψ(z,λ)e λz 1 as λ , for all z C.
− −
7→ ∈ ∩ → → ∞ ∈
¯
This∂-equationcombinedwithR.Novikov’sintegralequationpermitsustofind, start-
ing from the Dirichlet-to-Neumann mapping, firstly the boundary values ψ , secondly
bΩ
the ”∂¯-scattering data” b(λ), and thirdly ψ . (cid:12)
Ω (cid:12)
(cid:12)
Summarizing, the conductivity function(cid:12) σ is thus retrieved from the given Dirichlet-
Ω
to-Neumann data by means of the scheme: (cid:12)
(cid:12)
ddc√σ
¯
DN data ψ ∂ scattering data ψ .
→ bΩ → − → Ω → √σ Ω
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
Main result
We suppose that instead of C we have a smooth algebraic Riemann surface V in
C2, given by an equation V = z C2; P(z) = 0 , where P is a holomorphic polyno-
{ ∈ }
mial of degree d 1. Put z = w /w , z = w /w and suppose that the projective
1 1 0 2 2 0
compactification V˜≥ of V in CP2 C2 with coordinates w = (w : w : w ) intersects
0 1 2
⊃
CP1 = z CP2; w = 0 transversally in d points. In order to extend the Novikov
0
reco∞nstru{ctio∈n scheme on the}Riemann surface V C2 we need, firstly, to find an appro-
¯ ⊂
priate Faddeev type Green function for ∂(∂+λdz ) on V. One can check that for the case
1
V = C the Faddeev–Green function g(z,λ) is a composition of Cauchy–Green–Pompeiu
¯
kernels for the operators f ϕ = ∂f and u f = (∂ + λdz)u, where u, f, and ϕ are
7→ 7→
respectively a function, a (1,0)-form, and a (1,1)-form on C. More precisely, one has the
formula
1 eλw λ¯w¯dw dw¯
−
g(z,λ) = − ∧ .
i(2π)2 Z (w+z) w¯
C ·
w
∈
2
The main purpose of this paper is to construct an analogue of the Faddeev–Green function
ˆ
on the Riemann surface V. To do this we need to find explicit formulas f = Rϕ and
¯
u = R f (with appropriate estimates), for solutions of the two equations ∂f = ϕ and
λ
(∂ +λdz )u = f f on V. Here we consider V equipped with the Euclidean volume
1 λ
−H
form ddc z 2, and we require ϕ L1 (V), f W˜ 1,p˜(V), and u L (V), p˜ > 2, with
| | ∈ 1,1 ∈ 1,0 ∈ ∞
f being the projection of f on the subspace of pseudoholomorphic (1,0)-forms on V˜:
λ
H¯ ¯
∂ f = λdz¯ f.
λ 1 λ
H ∧H
¯
The new formulas obtained in this paper for solution of ∂f = ϕ and (∂ +λdz )u = f
1
on V one can interprete as explicit and more precise versions of the classical Hodge–
Riemann decomposition results on Riemann surfaces. We will define the Faddeev type
Green function for ∂¯(∂+λdz ) on V as the kernel g (z,ξ) of the integral operator R Rˆ.
1 λ λ
◦
Further results
Let σ C(2)(V), with σ > 0 on V, and σ const on a neighborhood of V˜ V. Let
a ,...,a b∈e generic points in this neighborhood≡, with g being the genus of V˜. U\sing the
1 g
FaddeevtypeGreenfunctionconstructed here, wehavein[HM]obtainednaturalanalogues
of all steps of the Novikov reconstruction scheme on the Riemann surface V. In particular,
under a smallness assumption on dlog√σ, the existence (and uniqueness) of the solution
µ(z,λ) of the Faddeev type integral equation
g
i µ(ξ,λ)ddc√σ
µ(z,λ) = 1+ g (z,ξ) +i c g (z,a ), z V, λ C
λ l λ l
2 Z √σ ∈ ∈
X
l=1
ξ V
∈
holds for any a priori fixed constants c ,...,c . However (and this was overlooked in
1 g
[HM]), there exists only one unique choice of constants c = c (λ,σ) for which the integral
l l
equation above is equivalent to the differential equation
g
i ddc√σ
¯
∂(∂ +λdz )µ = µ +i c δ(z,a ),
1 l l
2 √σ
(cid:0) (cid:1) Xl=1
where δ(z,a ) are Dirac measures concentrated in the points a (see also 4 below).
l l
§
1. A Cauchy–Pompeiu type formula on an affine algebraic Riemann surface
§
By L (V) we denote the space of (p,q)-forms on V with coefficients in distributions
p,q
of measure type on V. By Ls (V) we denote the space of (p,q)-forms on V with absolutely
p,q
integrable in degree s 1 coefficients with respect to the Euclidean volume form on V.
If V = C and f is a fu≥nction from L1(C) such that ∂¯f L (C), then the generalized
0,1
∈
Cauchy formula has the following form
¯
1 ∂f(ξ) dξ
f(z) = ∧ , z C.
−2πi Z ξ z ∈
C −
ξ
∈
3
This formula becomes the classical Cauchy formula, when f = 0 on C Ω and f (Ω),
\ ∈ O
where Ω is some bounded domain with rectifiable boundary in C. The generalized Cauchy
formula was discovered by Pompeiu [P1] in connection with his solution of the Painlev´e
problem, i.e., in proving the existence for a totally disconnected compact set E with
positive Lebesgue measure of a non-zero function f (C E) C(C) L1(C). The
∈ O \ ∩ ∩
Cauchy–Pompeiu formula has a large number of fundamental applications: in the the-
ory of distributions (L.Schwartz), in approximations problems (E.Bishop, S.Mergelyan,
A.Vitushkin), in the solution of the corona problem (L.Carleson), in the theory of pseudo-
analytic functions (L.Bers, I.Vekua), and in inverse scattering and integrable equations
(R.Beals, R.Coifman, M.Ablowitz, D.Bar Yaacov, A.Fokas).
Motivated by applications to Electrical Impedance Tomography we develop in this
paper the Cauchy–Pompeiu type formulas on affine algebraic Riemann surfaces V C2
⊂
and give some applications.
Let V˜ be a smooth algebraic curve in CP2 given by the equation
V˜ = w CP2; P˜(w) = 0 ,
{ ∈ }
with P˜ being a homogeneous holomorphic polynomial in the homogeneous coordinates
w = (w : w : w ) CP2. Without loss of generality we may suppose that
0 1 2
∈
i) V˜ intersects CP1 = w CP2; w = 0 transversally, V˜ CP1 = a ,...,a ,
0 1 d
d = degP˜; ∞ { ∈ } ∩ ∞ { }
ii) V = V˜ CP1 is a connected curve in C2 with equation V = z C2; P(z) = 0 ,
where P\(z)∞= P˜(1,z ,z ) such that { ∈ }
1 2
∂P/∂z
1
const(V), if z r = const(V);
1 0
(cid:12)∂P/∂z (cid:12) ≤ | | ≥
(cid:12) 2(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
iii) For any z V, such that ∂P (z ) = 0 we have ∂2P(z ) = 0.
∗ ∈ ∂z2 ∗ ∂z22 ∗ 6
By the Hurwitz–Riemann theorem the number of such ramification points is equal to
d(d 1). Let us equip V with the Euclidean volume form ddc z 2.
− | |
Notation
Let W˜ 1,p˜(V) = F L (V);∂¯F Lp˜ (V) , p˜ > 2. Let us denote by Hp (V) the
p { ∈ ∞ ∈ 0,1 } 0,1
subspace in L (V), 1 < p < 2, consisting of antiholomorphic forms. For all p (1,2),
p 0,1 ∈
the space H (V) coincides with the space of antiholomorphic forms on V admitting an
0,1
antiholomorphic extension to the compactification V˜ CP2. Hence, by the Riemann–
p ⊂
Clebsch theorem one has dimCH (V) = (d 1)(d 2)/2 for all p (1,2).
0,1 − − ∈
Proposition 1. Let V be the connected components of z V; z > r . Then for
j 1 0
{ } { ∈ | | }
all j 1,...,d there exist operators R : Lp (V) Lp˜(V) and R : Lp (V) W˜ 1,p˜(V)
∈ { p } p 1 0,1 → 0 0,1 → p
and : L (V) H (V), 1 < p < 2, 1/p˜ = 1/p 1/2 such that, for all Φ L (V),
H 0,1 → 0,1 − ∈ 0,1
one has the decomposition
¯
Φ = ∂RΦ+ Φ, where R = R +R , (1.1)
1 0
H
4
¯
1 dξ ∂P ξ z¯
1
R Φ = Φ(ξ) det (ξ), − ,
1 2πi Z ∂P (cid:20) ∂ξ ξ z 2(cid:21)
ξ V ∂ξ2 | − |
∈ (1.2)
g
Φ = Φ ω ω¯ ,
j j
H Z ∧
Xj=1(cid:0) (cid:1)
V
with ω being an orthonormal basis for the holomorphic (1,0)-forms on V˜, i.e.,
j
{ }
ω ω¯ = δ , j,k = 1,2,...,g,
j k jk
Z ∧
V
and
lim RΦ(z) = 0.
zz∈Vj
→∞
Remark 1. If p [1,2) and q (2, ] the condition Φ Lp (V) Lq (V) implies that
∈ ∈ ∞ ∈ 0,1 ∩ 0,1
RΦ C(V˜).
∈
Remark 2. For the case when V = C = z C2; z = 0 Proposition 1 and Remark 1 are
2
{ ∈ }
reduced to the classical results of Pompeiu [P1], [P2] and of Vekua [V].
Remark 3. Based on the technique of [HP] one can construct an explicit formula not only
for the main part R of the R-operator, but for the whole operator R.
1
Proof of Proposition 1: Let Q(ξ,z) = Q (ξ,z),Q (ξ,z) be a pair of holomorphic poly-
1 2
{ }
nomials in the variables ξ = (ξ ,ξ ) and z = (z ,z ), such that
1 2 1 2
∂P
Q(ξ,ξ)= (ξ) and
∂ξ
(1.3)
def
P(ξ) P(z) = Q (ξ,z)(ξ z )+Q (ξ,z)(ξ z ) = Q(ξ,z),ξ z .
1 1 1 2 2 2
− − − h − i
The conditions i) and ii) imply that for ε small enough there exists a holomorphic retrac-
0
tion z r(z) of the domain = z C2 : P(z) < ε onto the curve V.
→ Uε0 { ∈ | | 0}
Put = z C2; P(z) < ε, z < r , where 0 < ε ε and r r . Put
ε,r 1 0 0
U { ∈ | | | | } ≤ ≥
also Vc = z C2; P(z) = c , where c C, c ε and Φ˜(z) = Φ(r(z)), z .
{ ∈ p } ∈ | | ≤ 0 ∈ Uε0
The condition Φ L (V) and properties of the retraction z r(z) together imply that
∈ 0,1 →
∂¯Φ˜ = 0 on and
Uε0
˜
Φ const(V) Φ , (1.4)
Lp(Vc) Lp(V)
k k ≤ ·k k
uniformly with respect to c, for c ε . By results from [H] and [Po] we can choose the
0
following explicit solution F˜ on| | ≤ of the ∂¯-equation ∂¯F˜ = Φ˜ :
ε,r Uε,r ε,r (cid:12)Uε,r
(cid:12)
5
¯ ¯
1 ξ z¯ ξ z¯
F˜ (z) = Φ˜ det − ,∂¯ − dξ dξ +
ε,r 2πi (cid:26) Z ∧ (cid:20) ξ z 2 ξ ξ z 2(cid:21)∧ 1 ∧ 2
(cid:0) (cid:1) | − | | − |
ξ∈Uε,r
¯
(ξ z¯ )
˜ 2 2
Φ − dξ dξ +
Z ∧(cid:20)− (ξ z ) ξ z 2(cid:21)∧ 1 ∧ 2 (1.5)
1 1
− | − |
ξ∈bUε,r: |ξ1|=r
¯
ξ z¯ Q
Φ˜ det − , dξ dξ , z .
Z ∧ (cid:20) ξ z 2 P(ξ) P(z)(cid:21)∧ 1 ∧ 2(cid:27) ∈ Uε,R
| − | −
ξ∈bUε,r: |P(ξ)|=ε
The property (1.4) implies that for any z V we have
∈
¯ ¯
ξ z¯ ξ z¯
Φ˜ det − ,∂¯ − dξ dξ 0, ε 0 and
Z ∧ (cid:20) ξ z 2 ξ ξ z 2(cid:21)∧ 1 ∧ 2 → →
| − | | − |
ξ∈Uε,r
¯
(ξ z¯ )
Φ˜ 2 − 2 dξ dξ 0, ε 0, r .
Z ∧(cid:20)− (ξ z ) ξ z 2(cid:21)∧ 1 ∧ 2 → → → ∞
1 1
− | − |
ξ∈bUε,r: |ξ1|=r
Hence for all z V there exists lim F˜ = F˜(z), where
ε,r
∈ ε 0
r→
→∞
¯
1 Φdξ ξ z¯
˜ 1
F(z) = det − ,Q(ξ,z) . (1.6)
−2πi Z ∂P (ξ) ∧ (cid:20) ξ z 2 (cid:21)
ξ V ∂ξ2 | − |
∈
From (1.5) and (1.6) it follows that
∂¯ F˜ = Φ(z). (1.7)
z V
(cid:12)
(cid:12)
Now put F = R Φ. Using (1.2), (1.3), (1.6), and (1.7) we obtain
1 1
¯
1 dξ ∂P ξ z¯
¯ 1 ¯
∂ F (z) = Φ(ξ) det (ξ),∂ − =
z 1 V 2πi Z ∧ ∂P ∧ (cid:20) ∂ξ z ξ z 2(cid:21)
(cid:12)(cid:12) ξ V ∂ξ2 | − |
∈
Φ+ 21πiξZV Φ(ξ)∧ d∂∂ξξP21 ∧ |ξ −1z|4 det(cid:12)(cid:12)(cid:12) ∂∂∂∂ξξPP12((ξξ)) −ξ(2ξ−1 −z2z1)(cid:12)(cid:12)(cid:12)det(cid:12)(cid:12)(cid:12)ξξ¯¯21 −−zz¯¯21 ddzz¯¯21(cid:12)(cid:12)(cid:12) =
∈ (cid:12) (cid:12) (cid:12) (cid:12)
Φ+KΦ,
where
KΦ = 1 Φ(ξ)∧dξ1 h∂∂Pξ (ξ),ξ−zi·h∂∂Pξ¯¯(z),ξ¯−z¯idz¯ . (1.8)
2πi Z ξ z 4 ∧ ∂P (ξ) ∂P¯(z) 1
ξ V | − | ∂ξ2 · ∂ξ¯2
∈
6
The estimate R Φ = F Lp˜(V) follows from the property Φ Lp(V) and the following
1 1
∈ ∈
estimate of the kernel for the operator R :
1
1 ¯
∂P − ∂P ξ z¯ 1
(ξ) det (ξ), − dξ = O ( dξ + dξ ),
(cid:12)(cid:18)∂ξ (cid:19) (cid:20) ∂ξ ξ z 2(cid:21) 1(cid:12) (cid:18) ξ z (cid:19) | 1| | 2|
(cid:12) 2 | − | (cid:12) | − |
(cid:12) (cid:12)
where ξ,z (cid:12) V. (cid:12)
∈
For the kernels of operators Φ KΦ and Φ ∂ KΦ we have the corresponding
z
7→ 7→
estimates
∂P(ξ),ξ z ∂P¯(z),ξ¯ z¯ dξ dz¯
h∂ξ − i·h∂ξ¯ − i 1 ∧ 1 =
(cid:12) ∂P (ξ) ξ z 4 ∂P¯(z) (cid:12)
(cid:12)(cid:12) ∂ξ2 ·| − | · ∂ξ¯2 (cid:12)(cid:12)
(cid:12) (cid:12) (1.9)
O 1 (dξ +dξ ) (dz¯ +dz¯ ) if ξ z 1,
1+z 2 | 1 2 ∧ 1 2 | | − | ≤
(cid:26)O(cid:0) 1| | (cid:1) (dξ +dξ ) (dz¯ +dz¯ ) if ξ z 1.
ξ z 2 | 1 2 ∧ 1 2 | | − | ≥
(cid:0)| − | (cid:1)
∂P(ξ),ξ z ∂P¯(z),ξ¯ z¯ dξ dz¯
∂ h∂ξ − i·h∂ξ¯ − i 1 ∧ 1 =
(cid:12) z ∂P (ξ) ξ z 4 ∂P¯(z) (cid:12)
(cid:12)(cid:12) ∂ξ2 ·| − | · ∂ξ¯2 (cid:12)(cid:12)
(cid:12) (cid:12)
(1.10)
O 1 (dξ +dξ ) (dz¯ +dz¯ ) (dz +dz ) if ξ z < 1,
(1+z 2)ξ z | 1 2 ∧ 1 2 ∧ 1 2 | | − |
 (cid:0) | | | − |(cid:1)
 O 1 (dξ +dξ ) (dz +dz ) (dz¯ +dz¯ ) if ξ z 1,
ξ z 3 | 1 2 ∧ 1 2 ∧ 1 2 | | − | ≥
(cid:0)| − | (cid:1)

ξ,z V.
∈
These estimates imply that, for all p˜> 2 and p > 1, one has
def 1,p˜ p
Φ = KΦ W (V) L (V). (1.11)
0 ∈ 0,1 ∩ 0,1
From estimates (1.9)-(1.11)it follows that the (0,1)-form Φ = KΦ on V can be considered
0
alsoasa(0,1)-formonthecompactificationV˜ ofV inCP2 belongingtothespaces W1,p(V˜)
0,1
for all p < 2, where V˜ is equipped with the projective volume form ddcln(1+ z 2).
| |
From the Hodge–Riemann decomposition theorem [Ho], [W] we have
¯ ¯
Φ = ∂(∂ GΦ )+ Φ , (1.12)
0 ∗ 0 0
H
where Φ H (V˜), and G is the Hodge–Green operator for the Laplacian ∂¯∂¯ +∂¯ ∂¯
0 0,1 ∗ ∗
H ∈
on V˜ with the properties
G(H (V˜)) = 0, ∂¯G = G∂¯, ∂¯ G = G∂¯ .
0,1 ∗ ∗
The decomposition (1.12) implies that
∂¯ GΦ W2,p(V˜), p (1,2) and Φ H (V˜),
∗ 0 0 0,1
∈ ∈ H ∈
7
and this in turn implies that ∂¯ GΦ C(V˜). Returning to the affine curve V with the
∗ 0
∈
Euclidean volume form, we obtain that
R˜ Φ d=ef ∂¯ GKΦ W˜ 1,p˜(V), p˜> 2, where
0 ∗
V ∈ ∀
(cid:12)
W˜ 1,p˜(V) d=ef F (cid:12) L (V); ∂¯F Lp˜ (V) , (1.13)
{ ∈ ∞ ∈ 0,1 }
def p
and Φ = KΦ H (V), p > 1.
H H V ∈ 0,1
(cid:12)
(cid:12)
Now put R˜ = R + R˜ . Then, for all Φ Lp (V), we have R˜ Φ W˜ 1,p˜(V), and
1 0 ∈ 0,1 0 ∈
R˜Φ L (V) Lp˜(V).
∞
∈ ∪
By Corollary 1.1 below, which is based only on (1.13), it follows that, for any form
p
Φ L (V), one has a limit
∈ 0,1
˜ def ˜
lim RΦ(z) = RΦ( ).
j
z ∞
z→∈V∞j
Put R Φ = R˜ Φ R˜Φ( ) and RΦ = R˜Φ R˜Φ( ). We then have property (1.1) for
0 0 j j
− ∞ − ∞
R = R +R . This concludes the proof of Proposition 1.
1 0
Corollary 1.1. LetF L (V)and∂¯F Lp (V), 1 < p < 2. Then, forallj 1,...,d ,
∈ ∞ ∈ 0,1 ∈ { }
there exists a limit lim F(z) d=ef F( ) such that (F F( )) Lp˜.
z ∞j − ∞j Vj ∈
z→∈V∞j (cid:12)(cid:12)
Proof: Put ∂¯F = Φ. Then by (1.13) we have R˜Φ L (V) Lp˜(V) and ∂¯(F R˜Φ) = Φ.
∞
˜ ∈ ∪ − H
Then the function h = F RΦ is harmonic on V. The estimates F L (V) and
∞
R˜Φ Lp˜(V) L (V) imply−by the Riemann extension theorem that h ca∈n be extended
∞
∈ ∪ ˜ ˜ ˜
to a harmonic function h on V. Hence, h = F RΦ const = c. This implies that there
− ≡
def
exists lim F(z) = c = F( ). Corollary 1.1 is proved.
j j
z ∞
z→∈V∞j
Corollary 1.1 admits the following useful reformulation.
Corollary 1.2. In the notations of Proposition 1, for any bounded function ψ on V, such
that ∂¯ψ Lp(V), 1 < p < 2, the following formula is valid:
∈
1 det ∂P(ξ),ξ¯ z¯ dξ
ψ(z) = ψ( )+R ∂¯ψ + ∂¯ ∂ξ − ψ,
∞j 0 2πi Z ξ(cid:18) ∂(cid:2)P (ξ) ξ z 2(cid:3) (cid:19)∧
ξ V ∂ξ2 ·| − |
∈
where R ∂¯ψ W˜ 1,p˜(V), 1/p˜= 1/2 1/p, and R ∂¯ψ(z) 0, for z V , with z .
0 0 j
∈ − → ∈ → ∞
2. Kernels and estimates for ∂¯f = ϕ with ϕ L1 (V)
§ ∈ 1,1
Let ϕ be a (1,1)-form of class L (V) with support in V = z V; z r , where r
∞1,1 0 { ∈ | 1| ≤ 0} 0
satisfies the condition ii) of 1.
§
8
If V = C then by classical results from [P1] and [V], the Cauchy–Pompeiu operator
dz ϕ(ξ)
ϕ d=ef Rˆϕ
7→ 2πi Z ξ z
−
ξ∈V0
determines a solution f = Rˆϕ for the equation ∂¯f = ϕ on C with the property
f W1,p˜(C) (C V ) for all p˜> 2.
∈ 1,0 ∩O1,0 \ 0
In this section we derive an analogous result for the case of an affine algebraic Riemann
surface V C2.
⊂
Let V V = d V , where V are the connected components of V V .
\ 0 ∪j=1 j { j} \ 0
Lemma 2.1. Let Φ = dz ϕ and f = Fdz = (RΦ)dz , where R is the operator from
1 1 1
⌋
Proposition 1. Then
p
i) Φ L (V ), p [1,2), Φ = 0 on V V ;
∈ 0,1 0 ∈ \ 0
ii) F W1,p(V ) p (1,2), f W˜ 1,p˜(V) p˜ (2, ),
V0 ∈ 0 ∀ ∈ ∈ 1,0 ∀ ∈ ∞
∂¯F(cid:12) = Φ Φ, where is the operator from Proposition 1,
(cid:12)
¯ −H H
∂f = ϕ ( Φ) dz , and
1
− H ∧
F + F const(V,p) Φ , 1/p˜= 1/p 1/2;
k kL∞(V\V0) k kLp˜(V0) ≤ k kLp(V) −
iii) if, in addition, ϕ W1, (V), then f W˜ 2,p˜(V).
∞
∈ ∈
Proof:
i) The property Φ = 0 follows from ϕ = 0.
Put V = z (cid:12)VV\;V0 ∂P ∂P . T(cid:12)hVe\Vd0efinition Φ = dz ϕ implies that
0± { ∈(cid:12) 0 ± ∂z2 ≥ ± ∂z1 } (cid:12) ⌋
(cid:12) (cid:12) (cid:12) (cid:12)
Φ =(cid:12)Φ+(cid:12)dz¯ , (cid:12)whe(cid:12)re Φ+ L (V+) and
V+ 1 ∈ ∞ 0
(cid:12) j (2.1)
Φ(cid:12) = Φ dz¯ /(∂z /∂z ), where Φ L (V ).
V− − 2 1 2 − ∈ ∞ 0−
(cid:12) j
(cid:12)
p
The properties (2.1) imply that Φ L (V ) for all p (1,2).
∈ 0,1 0 ∈
¯ ¯
ii) The equalities ∂F = Φ Φ and ∂f = ϕ ( Φ) dz follow from Proposition 1
1
− H − H ∧
together withthe definitions Φ = dz ϕ and f = Fdz . The inclusions F L (V V )
1 1 ∞ 0
⌋ ∈ \
and F W1,p(V ) follow from the formula F = RΦ and Proposition 1. The
0
V0 ∈
inclusio(cid:12)n f W˜ 1,p˜(V) follows from the equalities ∂¯f = ϕ ( Φ) dz , f = Fdz
(cid:12) ∈ 1,0 − H ∧ 1 1
and Proposition 1.
iii) If, in addition, ϕ W1,∞(V), then f W˜ 2,p˜(V). It follows from equalities above
∈ 1,1 ∈ 1,0
1,
with ϕ ∈ W1,1∞(V) and suppϕ ⊂ V0.
Lemma 2.2. For each (0,1)-form g Hp (V) there exists a (1,0)-form h Lp (V˜)
∈ 0,1 ∈ 1,0
˜
(1 p < 2), unique up to adding holomorphic (1,0)-forms on V, such that
≤
¯
∂h = gdz . (2.2)
V˜ 1
(cid:12)
(cid:12)
9
Proof: For any g Hp (V) the (1,1)-form g dz determines a current G on Y˜ by the
∈ 0,1 ∧ 1
equality
d
def
G,χ = lim (χ χ ( ))gdz +χ ( ) g dz ,
j 1 j 1
h i R Z − ∞ ∞ Z ∧
→∞Xj=1(cid:2) (cid:3)
Vj {z∈Vj; |z1|<r}
where χ C(ε)(V˜), ε > 0 and χ ( ) = lim χ(z).
j
∈ ∞ zz∈Vj
→∞
By Serre duality [S], the current G is ∂¯-exact on V˜ if and only if
G,1 = lim g dz = 0. (2.3)
1
h i R Z ∧
→∞
{z∈V; |z1|≤r}
Let us check (2.3). We have
g dz = z g.
1 1
Z ∧ − Z ∧
{z∈V: |z1|≤r} {z∈V: |z1|=r}
Putting w = 1/z into the right-hand side of this equality, we obtain
1 1
d
dw¯
1
g dz = g (w¯ ) = 0.
1 j 1
Z ∧ − Z w
X 1
j=1
{z∈V: |z1|≤r} |w1|=1/r
Here the last equality follows from the properties
g (w¯ )dw¯ = g and g¯ (D(0,1/r)).
j 1 1 (cid:12)Vj∩{|w1|≤1/r} j ∈ O
(cid:12)
Hence, by (2.3) there exists h L1 (V˜) such that equality (2.2) is valid in the sense of
∈ 1,0
currents. Moreover, any solution of (2.2) automatically belongs to Lp (V˜), 1 < p < 2.
1,0
Such a solution h of (2.2) is unique up to holomorphic (1,0)-forms on V˜ because the
conditions h Lp (V˜) and ∂¯h = 0 on V imply that h extends as a holomorphic (1,0)-form
∈ 1,0
on V˜.
Notation: Let :Hp (V) Lp (V˜) (1 < p < 2) be the operator defined by the formula
H⊥ 0,1 → 1,0
g g, where g is the unique solution h of (2.2) in Lp (V˜) with the property
7→ H⊥ H⊥ 1,0
p
h g˜ = 0 for all g˜ H (V).
Z ∧ ∈ 0,1
V
10