H

DOI 10.4134/JKMS.2011.48.2.241

H ¨ OLDER ESTIMATES FOR THE CAUCHY-RIEMANN EQUATION ON PARAMETERS

Sanghyun Cho

Abstract. Let {Ωτ}τ∈I be a family of strictly convex domains inCⁿ. We obtain explicit estimates for the solution of the ∂-equation on Ω× I in H¨older space. We also obtain explicit point-wise derivative estimates for the ∂-equation both in space and parameter variables.

1. Introduction

Let I ⊂ R

be a bounded open set containing 0, and let {Ω

}

be a family of smoothly bounded domains in C

with smooth defining function ρ

for Ω

for each τ ∈ I, and set

Ω

:= ∪

τ∈I

Ω

× {τ} = Ω × I.

Then Ω

is a bounded domain in R

.

Definition 1.1. {Ω

}

is said to be a smooth strongly convex perturbation family of Ω

if Ω

is strongly convex for each τ ∈ I and there is a family of diffeomorphisms {Ψ

}

such that

(1) Ψ

: Ω

→ Ω

, Ψ

= Identity,

(2) Ψ

(bΩ

) = bΩ

for each τ ∈ I and Ψ

is smooth on τ ∈ I variable.

The solvability of the Cauchy-Riemann equation (∂-equation) and the tan- gential Cauchy-Riemann equation (∂

-equation) and the estimates of the solu- tions in various topologies such as in Sobolev, L

and H¨ older spaces are key subjects in several complex variables for last several decades [1, 6, 7, 8, 9, 10].

To study the local behavior of the solutions of ∂ or ¯ ∂

equation, we some- times need to construct a family of strongly pseudoconvex (or strongly convex) domains which are foliated inside of a domain near a boundary point. Let D ⊂ C

be a domain and bD is smooth near z

∈ bD and the Levi-form of

Received August 27, 2009; Revised November 28, 2009.

2010 Mathematics Subject Classification. 32W05, 32W10.

Key words and phrases. H¨older estimates, integral representation for ∂, Cauchy-Riemann equation.

Partially supported by KRF-2005-070-C00007 from KRF and by Sogang University.

⃝2011 The Korean Mathematical Societyc 241

bD has k positive eigenvalues at z

. In this situation, the author constructed a family of smooth strongly convex family of domains of complex dimension k which are foliated inside of D, making a neighborhood of z

in D, and got estimates of ∂ and ¯ ∂

equation in Sobolev spaces near z

∈ bD [3].

Let W be a set and k be a nonnegative integer and 0 < α ≤ 1. We say f ∈ C

(W ) if the norm defined by

|f|

:= |f|

+ |D

f |

is finite. Here |f|

denotes the C

norm on W and |D

f |

denotes the sum of the H¨ older norm of all derivatives of f of order k. We set C

= C

, and set C

= C

, the usual H¨ older space, and write |f|

= |f|. We can extend

| · |

to the (p, q)-forms, denoted by C

, by estimating its components.

In this paper, we prove stability of H¨ older estimates for ∂ and obtain point- wise derivative estimates for ∂-equation on Ω × I for space and parameter variables. Concerning the parameter dependence of the ∂-equation, one can refer to the most fundamental papers of Hamilton [5].

In the sequel, we let {Ω

}

be a smooth strongly convex perturbation family of Ω

⊂⊂ C

with smooth defining function ρ

for Ω

for each τ ∈ I.

We also assume that |I|, the diameter of I, is sufficiently small so that the estimate (2.4) in Section 2 holds. The following theorem is the stability of the H¨ older estimates.

Theorem 1.2. For any f

∈ C

(Ω

), 1 ≤ q ≤ n, such that ∂f

= 0 in Ω

in the distribution sense, there exists u

∈ C

(Ω

) such that ∂u

= f

in Ω

and

(1.1) |u

|

≤ C|f

|

, where C is a constant independent of f

and τ ∈ I.

When we study function theories on domains in C

, such as extending ¯ ∂

- closed forms on bD to ∂-closed forms on D with H¨ older or Sobolev estimates, we sometimes need to get derivative estimates for the solutions of ∂-equation on Ω

in space as well as in parameter variables. Let D

and D

be the differential operators of order l

in space variables and of order l

in parameter variables respectively. Also if f = ∑

|I|=p,|J|=q

f

dz

∧ d¯z

∈ C

, we set

|D

f (z) | := ∑

|I|=p,|J|=q

|D

f

(z) |,

where |D

f

(z) | = ∑

|α|=l

|D

f

(z) | and α is a multi-index.

We have the following derivative estimates for the ∂-equation in H¨ older space. As for the estimates in Sobolev space [3], we lose some regularity.

Theorem 1.3. Assume f

∈ C

(Ω

), 1 ≤ q ≤ n, 0 < α < 1, and ∂f

= 0

(in distribution sense if k = 0) in Ω

. Then there exists u

∈ C

(Ω

)

such that ∂u

= f

in Ω

, and for each 0 < α

< α and for each z

∈ Ω

, we have

(1.2)

|D

u

(z

) | + |D

u

(z

) |

≤ C

( |ρ

(z

) |

∑

0≤j≤l−1

|D

f

(z

) | + |f

|

+ |ρ

(z

) |

|f

|

+ |ρ

(z

) |

|f

(z

) | ) for 1 ≤ l ≤ k + 1, where C

is a constant independent of τ ∈ I and f

. Remark 1.4. In Theorem 1.2, we gain 1/2 derivatives. We note that

|ρ(z)|

|f|

is more or less same as |f|

in a sense. Therefore we may regard the third term in the right hand side of (1.2) is equivalent to |f|

. Assuming that 0 < α ≤ 1/2, we therefore gain 1/2 + α

/2 − α derivatives which is smaller than 1/2 because α > α

.

In some cases, f is given in the form f = ρ

g for some k ≥ 1, 0 < α < 1 and for some form g. In this case the estimate in (1.2) becomes:

Corollary 1.5. Under the same assumptions as in Theorem 1.3, we further assume that D

f

= 0 on bΩ

, for each 0 ≤ l ≤ k − 1, for some k ≥ 1. Then (1.3) |D

u

(z

) | + |D

u

(z

) | ≤ C

|f

|

.

Remark 1.6. (1) When k = 1, (1.3) is the derivative estimate of the solution both on space and parameter variables.

(2) Let D be a smoothly bounded domain in C

with smooth defining func- tion ρ, and let h

be a ∂

-closed form on bD. To solve ∂-closed extension problem on D with H¨ older estimates, we first extend h

to h defined on D so that f := ∂h = ρ

g for some k ≥ 1, 0 < α < 1 and for some form g, depending on the regularity of h

.

2. Integral representation on parameter

Let Ω

be a smoothly bounded strongly convex domain in C

and let {Ω

}

be a smooth strongly convex perturbation family of Ω

with smooth defining function ρ

for Ω

for each τ ∈ I. In this section, we consider the solutions of ∂-equation in H¨ older space in space and parameter variables and prove the stability of the solution in H¨ older space as well as pointwise deriv- ative estimates of the solution. In the sequel, we let A ≲ B mean that there is a constant C, independent of parameter τ , such that A ≤ CB (C may be different from line to line).

Definition 2.1. Let D be a domain in C

with smooth boundary bD. A

C

function G(ζ, z) = (g

(ζ, z), . . . , g

(ζ, z)) is called a Leray map for D if it

satisfies ⟨G(ζ, z), ζ − z⟩ ̸= 0 for every (ζ, z) ∈ bD × D.

For each τ ∈ I, consider the map G

(ζ, z) :=

( ∂ρ

∂ζ )

= ( ∂ρ

∂ζ

, . . . , ∂ρ

∂ζ

)

,

where ρ

= ρ

(ζ) is a boundary defining function for Ω

. Then G

is indepen- dent of z, and it is a Leray map because Ω

is a strongly convex domain for each τ ∈ I. Using G

, we set

Φ

= ( 1

2πi

)

⟨ ζ ¯ − ¯z, dζ ⟩

|ζ − z|

∧

(⟨ d¯ ζ − d¯z, dζ ⟩

|ζ − z|

)

,

Φ

= ( 1

2πi )

⟨

G

, dζ ⟩

⟨G

, ζ − z⟩ ∧

(⟨ ∂ ¯

G

, dζ ⟩

⟨G

, ζ − z⟩

)

, (2.1)

and

Φ

= ( 1

2πi

)

⟨ ζ ¯ − ¯z, dζ ⟩

|ζ − z|

∧

⟨ G

, dζ ⟩

⟨G

, ζ − z⟩

∧ ∑

k1+k2=n−2

(⟨ d¯ ζ − d¯z, dζ ⟩

|ζ − z|

)

∧

(⟨ ∂ ¯

G

, dζ ⟩

⟨G

, ζ − z⟩

)

, (2.2)

where ¯ ∂

= ¯ ∂

+ ¯ ∂

. We use the notation Φ

, Φ

and Φ

to denote the summand of forms with degree (0, q) in z in Φ

, Φ

and Φ

respectively.

Note that Φ

is independent of τ and Φ

and Φ

depend on G

which is the derivatives of ρ

.

Since Ω

⊂ C

is strongly convex, the denominators of Φ

and Φ

are different from zero for ζ ∈ bΩ

and z ∈ Ω

, and they depend smoothly on τ . We have the following homotopy formula for ¯ ∂ on convex domains by Leray- Koppelman.

Theorem 2.2. For f

∈ C

( ¯ Ω

), 1 ≤ q ≤ n, we have f

(z) = ¯ ∂

T

f

(z) + T

∂f ¯

(z), z ∈ Ω

, where

(2.3) T

f

(z) =

∫

Ωτ

Φ

(ζ, z) ∧ f

(ζ) −

∫

bΩτ

Φ

(ζ, z) ∧ f

(ζ).

Note that ρ

−→ ρ

in C

topology as τ → 0 and there is a family of diffeomorphisms Ψ

: Ω

→ Ω

, Ψ(bΩ

) = bΩ

, Ψ

is identity and Ψ

is smooth on τ ∈ I variable. Since Ω

is strongly convex and ρ

= (Ψ

)

ρ

, there is a uniform constant c > 0, independent of τ ∈ I, such that

(2.4) c |a|

≤

∑

∂

ρ

∂x

∂x

(x)a

a

for all x ∈ bΩ

and a ∈ R

provided |I|, the diameter of I, is sufficiently small.

The estimate in (2.4) is essential in the estimates of Lemma 11.2.9 in [2] and we obtain that

Lemma 2.3. There is a constant c, independent of τ ∈ I, such that for any ζ ∈ bΩ

, and z ∈ Ω

,

(2.5) Re ⟨

G

, ζ − z ⟩

≥ c (

−ρ

(z) + |ζ − z|

) . Set

h

(ζ, z) = ⟨G

, ζ − z⟩ and h

(ζ, z) = |ζ − z|

.

As in the proof of Lemma 11.2.10 in [2], for each fixed z

near bΩ

, there is a special coordinate system (t, y) = (t

, t

, y) defined in a neighborhood U of z

such that t

(z

) = 0, and y = ρ

(ζ), t

= Imh

(ζ, z). Then the estimate in (2.5) shows that there exist constants c

> 0 and C

independent of τ ∈ I such that

(2.6) h

(ζ, z) ≥ c

( |ρ

(z) | + |t

|

+ |t

|), c

( |ρ

(z) | + |t|) ≤ |ζ − z| ≤ C

( |ρ

(z) | + |t|).

The estimates in (2.6) are the key ingredients to prove the

-H¨ older estimates for ∂ on strictly convex domains.

Let {Ψ

}

be the family of diffeomorphisms defined before (2.4) or in the Definition 1.1. By the generalized change of variables theorem, one obtains that

(2.7)

∫

Ωτ

ω =

∫

Ω

Ψ

ω, and

∫

bΩτ

ω

=

∫

bΩ

Ψ

ω

,

where ω and ω

are 2n and 2n − 1 forms on Ω

and bΩ

respectively. Using the relations in (2.7), we may bring the estimates on Ω

onto Ω

.

For any Φ

∈ C

(Ω

), set Φ

= Ψ

Φ

∈ C

(Ω

). Since |D

Ψ

| ≤ ˜ C

, independent of τ ∈ I, it follows that

(2.8) |D

Φ

(z

) | ≤ C

∑

|D

Φ

(z) |

for an independent constant C

> 0, where z

= Ψ

(z).

Note that the estimates in (2.5) and (2.6) are the key ingredients to prove the H¨ older estimates of ∂-equation gaining

derivative. Since the estimates in (2.5) and (2.6) are independent of τ ∈ I, and the solution operator in (2.3) is the Bochner-Martinelli integral, we can pull-back the estimates of the solution of

∂-equation on Ω

to the estimates of the solution on Ω

via (2.7) and (2.8) with

k = 1. Then the rest of the proof will be the same as in the proof of Theorem

11.2.11 in [2] proving Theorem 1.2, the stability of the H¨ older estimates.

3. Derivative estimates

In this section, we want to get derivative estimates for the solutions of ∂- equation on Ω

in space as well as in parameter variables. As for the estimates in Sobolev space [3], we lose some regularity when we take derivatives of the solution. For each z, ζ ∈ Ω

, we set z

= Ψ

(z) and ζ

= Ψ

(ζ), and we let ρ

(z) = ρ(Ψ

(z)).

Lemma 3.1. Assume g

∈ C

(Ω

) and set g = Ψ

g

. Then

(3.1) |D

g(z) | ≲

∑

|D

g

(z

) | and |D

g

(z

) | ≲

∑

|D

g(z) |.

Proof. Note that Ψ

→ I

as τ → 0 in C

(Ω × I) topology, where I

is the identity map on Ω

. Therefore it follows that |D

[Ψ

(z)

] | ≤ C

for any integers m, p ≤ k. Since D

g(z) is the sum of the terms of the form D

g

(z

)D

[Ψ

(z)

], where 1 ≤ l ≤ k and m, p ≤ k, the first estimate in (3.1) holds. Similarly, if we consider the map Ψ

, the second estimate in (3.1)

holds. □

In the sequel, we let β

(z) be a ball of radius ϵ > 0 with center at z. We need the following integral estimates for the form Φ

defined in (2.1). We note that Φ

is independent of τ ∈ I.

Lemma 3.2. Let D be a smoothly bounded domain in C

with smooth defining function ρ. For each z ∈ D and for any ϵ > 0 such that β

(z) ⊂ D, we set D

= D \ β

(z). Then for any α

> 0 we have

(3.2)

∫

bD

|Φ

(ζ, z) ||ζ − z|

≤ C

and

(3.3) 

 ∫

D_ϵ

∇

Φ

(ζ, z) 

 ≤ C

|ρ(z)|

.

Proof. Choose a neighborhood U of z where the special local coordinate func- tions (t, ρ) = (t

, t

, ρ) are defined where t = (t

, t

) are tangential coor- dinates. Using the expression of Φ

in (2.1), and integrating with respect to t

, and then using polar coordinates |t

| = r, we have

∫

bD∩U

|Φ

(ζ, z) ||ζ − z|

≲

∫

bD∩U

|ζ − z|

≲

∫

|t|≤A

dt

dt

· · · dt

( |t

| + |t

|)|t

|

≲

∫

r

| ln r|dr

r

≲ ˜ C

.

This estimate holds for each coordinate neighborhood and hence it follows that

∫

bD

|Φ

(ζ, z) ||ζ − z|

≤ C

. This proves (3.2).

To prove (3.3), we use Stokes’ theorem and then polar coordinates |ζ−z| = r:

  ∫

Dϵ

∇

Φ

(ζ, z)   ≤ ∫

|ζ−z|≥|ρ(z)|

|∇

Φ

(ζ, z) | +   

∫

ϵ≤|ζ−z|≤|ρ(z)|

∇

Φ

(ζ, z)   

≲

∫

|ρ(z)|

r

dr +

∫

|ζ−z|=ϵ

|Φ

(ζ, z) | +

∫

|ζ−z|=|ρ(z)|

|Φ

(ζ, z) |

≲ |ρ(z)|

,

because |ρ(z)|

ln |ρ(z)| ≲ ˜ C

for each α

> 0, and ϵ ≲ |ρ(z)|. □ To get the derivative estimates of ∂-equation, we need to get derivative estimates of Φ

. For a convenience, we drop the index τ in the expressions of z

, ζ

, ρ

, h

and Φ

etc. Then we have

(3.4)

D

Φ

(ζ, z)

≲

n

∑



 ∑

j=1

|ζ − z|

|h

|

|ζ − z|

+

l−1

∑

j=1

1

|h

|

|ζ − z|





:=

n

∑



 ∑

j=1

R

(ζ, z) +

l−1

∑

j=1

R

(ζ, z)



 := R

(ζ, z),

where h

= h

(ζ, z) satisfies the estimate (2.6).

Lemma 3.3. For each real numbers 0 < α

< α < 1, we have

∫

bΩ

R

(ζ, z) |ζ − z|

≤ C

|ρ(z)|

, and

∫

bΩ

R

(ζ, z) ≤ C|ρ(z)|

. (3.5)

Proof. Fix z ∈ Ω for a moment and choose a neighborhood U of z where the special local coordinate functions (t, y) = (t

, t

, y) are defined satisfying the estimates (2.6). Set δ = |ρ(z)| for a convenience. Let’s estimate, for example, the integral containing R

term which is the optimal case:

∫

bΩ∩U

R

(ζ, z) |ζ − z|

=

∫

bΩ∩U

1

|h

|

|ζ − z|

≤ C

∫

|t|≤A

dt

dt

· · · dt

(δ + |t

| + |t

|

)

(δ + |t

|)

(3.6)

for some constants A and C independent of τ , where we have used the estimates (2.6) and the fact that 2k − 1 − α ≥ 0. If we integrate the right side of (3.6) with respect to t

and then use polar coordinates |t

| = r, we obtain that

∫

bΩ∩U

R

(ζ, z) |ζ − z|

≲

∫

r

dr

(δ + r

)

(δ + r)

≲

∫

r

dr

δ

r

≤ C

δ

. (3.7)

Similarly the other integrals, involving R

, can be estimated and we get the first part of the estimate in (3.5).

For the second part of the estimate in (3.5), one obtains, as in (3.7), that

∫

bΩ∩U

R

(ζ, z) ≲

∫

r

dr

(δ + r

)

(δ + r)

≲

∫

dr

(δ + r

)δ

≤ Cδ

.

□ Now we are ready to prove Theorem 1.3, the pointwise derivative estimates for the solution of ∂-equation.

Proof of Theorem 1.3. We first assume that f

∈ C

(Ω

). Since |D

Ψ

| ≤ C

, independent of τ ∈ I, and by virtue of (2.7) and (2.8), we only need to estimate |D

u

(z

) |. Note that the solution u

is given by u

= T

f

because

∂f

= 0 where T

is given in (2.3). Set f = Ψ

f

, u = (Ψ

)

u

, and set z = Ψ

(z

). As in (2.7), we pull-back our solution to Ω

and we can write:

u(z) =

∫

Ω₀

Φ

(Ψ

(ζ), Ψ

(z)) ∧ f(Ψ

(ζ))

−

∫

bΩ0

Φ

(Ψ

(ζ), Ψ

(z)) ∧ f(Ψ

(ζ)) := u

(z) − u

(z).

(3.8)

In view of Lemma 3.1, it is enough to estimate |D

u(z) |.

At first, let’s estimate the derivatives of u

(z). For a convenience of notation, we set Ω

= Ω, ζ

= Ψ

(ζ) = ζ and z

= Ψ

(z) = z for each z, ζ ∈ Ω. For a moment, we fix z ∈ Ω and choose ϵ > 0 so that β

(z) ⊂ Ω, and set Ω

= Ω \β

(z) where β

(z) is the ball of radius ϵ > 0 with center at z. Then we can write (3.9) u

(z) =

∫

Ω_ϵ

Φ

(ζ, z) ∧ f(ζ) +

∫

β_ϵ(z)

Φ

(ζ, z) ∧ f(ζ) := I

(z) + I

(z).

Assume that f = f

dζ

, where J = (1, . . . , q) and set c

=

. Using the relation

j

(

|ζ−z|²ⁿ⁻²

)

= −2(n − 1)

, and by using Stokes’ theorem,

it follows that (3.10)

I

(z)

= c

∑

( −1)

n − 1

∫

βϵ(z)

∂

∂ζ

( 1

|ζ − z|

)

f

(ζ) ∧ d¯z

= c

∑

( −1)

n − 1

(∫

bβ_ϵ(z)

1

|ζ − z|

f

(ζ)[ ˆ dζ

] +

∫

β_ϵ(z)

1

|ζ − z|

∂f

∂ζ

(ζ) )

∧ d¯z

,

where ˆ. denotes the corresponding term is omitted. Similar expression holds for general (0, q)-form f . Therefore it follows that

(3.11)

|D

I

(z) | ≲

∫

bβ_ϵ(z)

|D

( |ζ − z|

) ||f(ζ)|+

∫

β_ϵ(z)

|D

( |ζ − z|

) || ∂f

∂ζ

(ζ) |

≲ |f| + O(ϵ)|f|

.

To estimate |D

I

(z) | for l ≥ 2, we use the expression of I

(z) in (3.10), and use Stokes’ theorem inductively together the fact that D

( |ζ − z|

) =

−D

( |ζ − z|

). Then one obtains that

(3.12) |D

I

(z) | ≤ C (|f|

+ O(ϵ)|f|

) .

Now let’s estimate D

I

(z). Note that the integrand of I

(z) is in C

class (in z-variable) and hence we can differentiate I

(z) under the integral sign. Again we use Stokes’ theorem inductively together the fact that D

( |ζ − z |

) = −D

( |ζ − z|

) to obtain that

|D

I

(z) | ≲   ∫

Ωϵ

∇

Φ

(ζ, z) ∧ D

f (ζ)   + 

 ∫

bΩ

Φ

(ζ, z) ∧ D

f (ζ)   +

∫

bβϵ(z)

Φ

(ζ, z) ∧ D

f (ζ)  

 := I

(z) + I

(z) + I

(z).

(3.13)

Note that

I

(z) ≤   

∫

bβ_ϵ(z)

Φ

(ζ, z) ∧ (

D

f (ζ) − D

f (z) )    + D

f (z) 

∫

bβϵ(z)

Φ

(ζ, z)  

 ≲ O (ϵ

) |f|

+ |D

f (z) |.

(3.14)

As in (3.14), we write I

(z) ≲

∫

Ω_ϵ

∇

Φ

(ζ, z) | D

f (ζ) − D

f (z) | + |D

f (z) | 

 ∫

Ωϵ

∇

Φ

(ζ, z) 

 := I

(z) + I

(z).

(3.15)

In view of (3.3), we have

(3.16) I

(z) ≤ C

|ρ(z)|

|D

f (z) | for each α

> 0.

Using polar coordinate |ζ − z| = r and from the fact that f ∈ C

(Ω), l − 1 ≤ k, it follows, from (2.1), that

(3.17)

I

(z) ≤ C|f|

∫

Ω

∇

Φ

(ζ, z) | ζ − z|

≤ ˜ C

|f|

∫

r

dr ≤ C

|f|

for some A > |Ω|, where |Ω| denotes the diameter of Ω. Combining (3.16) and (3.17), we obtain that

(3.18) I

(z) ≤ C

( |ρ(z)|

|D

f (z) | + |f|

)

for each α

> 0.

If we use the estimate (3.2), and use the method similar to the estimate (3.15), we obtain that

(3.19)

I

(z) ≲ |f|

+ |D

f (z) ||ρ(z)|

∫

bΩ

|Φ

(ζ, z) ||ζ − z|

≤ ˜ C

( |f|

+ |D

f (z) ||ρ(z)|

)

because |ρ(z)| ≲ |ζ − z| for ζ ∈ bΩ. Letting ϵ → 0, it follows, from (3.9) and (3.12)–(3.19), that for each 0 < α

< α < 1, we have

(3.20) |D

u

(z) | ≤ C

( |f|

+ |D

f (z) ||ρ(z)|

)

for l − 1 ≤ k and for f ∈ C

(Ω), where C

is independent of τ ∈ I.

By regularization process, we can approximate f ∈ C

(Ω) by a sequence of f

∈ C

(Ω) such that f

converges to f in C

(Ω) space, and the cor- responding solutions u

converges to u

in C

(Ω) space as δ converges to zero. Thus (3.20) holds for f ∈ C

(Ω). In view of Lemma 3.1, we see that

|D

u

(z

) | is bounded by the first two terms in the right hand side of (1.2).

Next we estimate |D

u

(z) |. For a convenience, we also drop the index τ in

the expressions of z

, ζ

, f

and ρ

etc. Choose a neighborhood U of z where

the special local coordinate functions (t, y) = (t

, t

, y) are defined satisfying

the estimates (2.6). Since u

is a smooth function, we can differentiate under the integral sign. Using (2.2), (2.6), (3.4) and (3.5), one obtains that

(3.21)

∫

bΩ∩U

 D

Φ

(ζ, z) ∧ f(ζ) 

≤

∫

bΩ∩U

 D

Φ

(ζ, z) | f (ζ) − f(z)| + |f(z)|

∫

bΩ∩U

 D

Φ

(ζ, z) 

≲ |f|

∫

bΩ∩U

R

(ζ, z) |ζ − z|

+ |f(z)|

∫

bΩ∩U

R

(ζ, z)

≲ |ρ(z)|

|f|

+ |ρ(z)|

|f(z)|

and hence it follows that (3.22) |D

u

(z) | ≤ C

( |f|

|ρ(z)|

+ |f(z)||ρ(z)|

) . If we combine (3.22) and Lemma 3.1, we see that (1.2) holds proving Theo-

rem 1.3. □

Lemma 3.4. For any real number α > 0, suppose f ∈ C

(D) and f = 0 on bD. Then

sup

z∈D

|ρ(z)|

|f(z)| ≤ C|f|

.

Proof. For each fixed z ∈ D, let ζ

be the projection of z onto bD. Then it follows that |z − ζ

| ≈ |ρ(z)|. Since f = 0 on bD, we have

|ρ(z)|

|f(z)| = |ρ(z)|

|f(z) − f(ζ

) | ≤ ˜ C

|ρ(z)|

|z − ζ

|

|f|

≤ C

|f|

.

□ Proof of Corollary 1.5. If we write u = u

+u

as in (3.8), it follows that u

= 0 because f

= 0 on bΩ

. By Theorem 1.3, Lemma 3.1 and Lemma 3.4, and from the given condition that D

f

= 0 on bΩ

for 0 ≤ l ≤ k − 1, it follows that (3.23) |D

f (z) ||ρ(z)|

≤ C

|f|

for 0 ≤ l ≤ k − 1. If we combine (3.1), (3.20) and (3.23), we see that (1.3)

holds and this proves Corollary 1.5. □

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Department of Mathematics Sogang University

Seoul 121-742, Korea

E-mail address: [email protected]