DSpace at VNU: L-p estimates for the (partial derivative)over-bar-equation on a class of infinite type domains

DSpace at VNU: L-p estimates for the (partial derivative)over-bar-equation on a class of infinite type domains tài liệu,...

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World Scientiﬁc Publishing Company

DOI: 10.1142/S0129167X14501067

L p estimates for the ¯∂-equation on a class

of infinite type domains

Ly Kim Ha

Department of Mathematics and Computer Science University of Science, Vietnam National University

227 Nguyen Van Cu Q.5

Ho Chi Minh City, Vietnam lkha@hcmus.edu.vn

Tran Vu Khanh

Tan Tao University, Tan Tao Avenue Tan Duc e-City, Long An Province, Vietnam Department of Mathematics National University of Singapore Blk S17, 10 Lower Kent Ridge Road Singapore 119076, Singapore khanh.tran@ttu.edu.vn mattvk@nus.edu.sg

Andrew Raich

Department of Mathematical Sciences SCEN 327, 1 University of Arkansas Fayetteville, AR 72701, USA araich@uark.edu

Received 1 October 2013 Accepted 21 October 2014 Published 18 November 2014

We proveL p estimates, 1≤ p ≤ ∞, for solutions to the Cauchy–Riemann equations

¯

∂u = φ on a class of inﬁnite type domains in C2 The domains under consideration are a

class of convex ellipsoids, and we show that ifφ is a ¯ ∂-closed (0, 1)-form with coeﬃcients

inL p andu is the Henkin kernel solution to ¯ ∂u = φ, then u p ≤ Cφ p where the constantC is independent of φ In particular, we prove L1 estimates and obtain L p

estimates by interpolation.

Keywords: ¯ ∂; Henkin solution; Henkin operator; L p estimates for ¯∂; inﬁnite type

domains.

Mathematics Subject Classiﬁcation 2010: 32W05, 32F32, 32T25, 32T99

Int J Math 2014.25 Downloaded from www.worldscientific.com by CORNELL UNIVERSITY on 12/17/14 For personal use only.

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1 Introduction

A fundamental question in several complex variables is to establish L p estimates

for solutions of the Cauchy–Riemann equation

¯

∂u = φ

on domains Ω ⊂ C n In this paper, we provide the ﬁrst examples of inﬁnite type

domains for which L p bounds hold, 1≤ p ≤ ∞ The domains under consideration

are a class of convex ellipsoids inC2, and we show that if φ is a ¯ ∂-closed (0, 1)-form

and u is the Henkin solution to ¯ ∂u = φ, then u p ≤ Cφ p where the constant C

is independent of φ Speciﬁcally, we prove L1 estimates and use the Riesz–Thorin

Interpolation Theorem to obtain L p estimates by interpolating with the L ∞

esti-mates established by Khanh [12] The L ∞estimates of Khanh generalized and were

inspired by the L ∞ estimates of Fornæss et al [9].

We investigate domains of the following form: Ω ⊂ C2 is a smooth, bounded

domain with a global deﬁning function ρ such that: for any p ∈ bΩ, there exist a

coordinates z p = T p (z) with the origin at p where T p is a linear transformation,

and functions F p , and r p such that

Ωp = T p(Ω) ={z p = (z p,1 , z p,2)∈ C2: ρ(T −1

p (z p )) = F p(|z p,1 |2) + r p (z p ) < 0 }

(1.1) or

Ωp = T p(Ω) ={z p = (z p,1 , z p,2)∈ C2

: ρ(z) = F p (x2p,1 ) + r p (z p ) < 0 } (1.2)

where z p,j = x p,j + iy p,j , x p,j , y p,j ∈ R, j = 1, 2, and i = √ −1 We also assume that

the functions F p:R → R and r p:C2→ R satisfy:

(1) F p(0) = 0;

(2) F

p (t), F p (t), F p (t), and ( F p t(t)) are non-negative on (0, d p);

(3) r p(0) = 0 and ∂r p

∂z p,2 = 0 on bΩ with |z1,p | ≤ δ;

(4) r pis convex,

where d p is the square of the diameter of Ωp and δ is a small number We will also

require that each F p satisﬁes a certain log integrability property.

This class of domains includes two well-known examples If Ω is of ﬁnite type

2m, then F p (t) = t m at the points of type 2m On the other hand, if F p (t) =

exp(−1/t α ), then Ω is of inﬁnite type at p, and this is our main case of interest We

call our domains Ω ellipsoids because they are generalizations of real and complex ellipsoids in C2 Classically, a complex ellipsoid in Cn is a domain of the form {z = (z1, , z n) ∈ C n : n

j=1 |z j |2m j < 1 }, and a real ellipsoid is a domain of

the form {z = (x1+ iy1, , x n + iy n) ∈ C n : n

j=1 (x2n j + y2m j ) < 1 } where

m j , n j ∈ N, 1 ≤ j ≤ n Our hypotheses include the following two classes of inﬁnite

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type domains:

Ω =





z = (z1, z2)∈ C2:

2

j=1

exp

− 1

|z j | α j

≤ e −1





; and

Ω =





z = (x1+ iy1, x2+ iy2)∈ C2:

2

j=1

exp

− |x1

j | α j

+ exp

− |y1

j | β j

≤ e −1





,

where α j , β j ∈ (0, 1) Moreover, our setting also includes domain

Ω =

z = (x1+ iy1, x2+ iy2)∈ C2: exp

1− |x1

1| α

+ χ(y1) +|z2|2≤ 1

where χ is a convex function and χ(y1) = 0 when|y1| < δ and 0 < α < 1 This is a

tube domain of inﬁnite type at 0 This domain provides an interesting example for studying necessary and suﬃcient conditions for local regularity of ¯∂ on domains of

inﬁnite type In fact, the canonical solution of ¯∂ associated to this tube domain has

superlogarithmic estimate and is locally regular if and only if α < 1 (see [1,6,14])

There is a long history of proving L pestimates for the ¯∂-equation, dating back

to the work of Kerzman [11] and Øvrelid [16] In [15], Krantz proved essentially

optimal Lipschitz and L p estimates on strongly pseudoconvex domains In the case

that Ω is a real ellipsoid, Diederich et al obtained sharp H¨older estimates [7] while

Chen et al established optimal L p estimates for complex ellipsoids [4] See also

Range [17] and Bruna and del Castillo [2] Both real and complex ellipsoids are domains of ﬁnite type, and the analysis in the referenced works depends in an essential fashion on the type In C2, Chang et al [3] proved L p estimates for the

¯

∂-Neumann operator on weakly pseudoconvex domains of ﬁnite type See [4,9] and the references within for a more complete history

More recently, there has been work on supnorm estimates for the Cauchy– Riemann equations on inﬁnite type domains in C2 Fornæss et al provided the

ﬁrst examples in [9] and Khanh found that the estimates hold when domains are

of the type (1.1) or (1.2) [12] In particular, Khanh proved the following

Theorem 1.1 ([12, Theorem 1.2]) If there exists δ > 0 so that

(1) Ω is defined by (1.1) and δ

0 |log F p (t2)| dt < ∞ for all p ∈ bΩ, or

0 |log (t) log F p (t2)| dt < ∞ for all p ∈ bΩ, then for any bounded, ¯ ∂-closed (0, 1)-form φ on Ω, the Henkin solution u on Ω satisfies ¯ ∂u = φ and

u L ∞(Ω)≤ Cφ L ∞(Ω), where C > 0 is independent of φ.

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In this paper, we will prove the L p-version of Theorem1.1 Our technique yields

L p-estimates both in the ﬁnite and inﬁnite type cases.

Theorem 1.2 If there exists δ > 0 and either of the following conditions hold :

0 |log F p (t2)| dt < ∞ for all p ∈ bΩ,

0 |log(t) log F p (t2)| dt < ∞ for all p ∈ bΩ, then for any ¯ ∂-closed (0, 1)-form φ in L p (Ω) with 1 ≤ p ≤ ∞, the Henkin kernel solution u on Ω satisfies ¯ ∂u = φ and

u L p(Ω)≤ Cφ L p(Ω), where C > 0 is independent of φ.

The following examples show that the L p estimates in Theorem 1.2 are sharp

in the case of inﬁnite type case

Example 1.1 For 0 < α < 1, let Ω be deﬁned by

Ω ={(z1, z2)∈ C2

: e1− 1

|z1|α +|z2|2

< 1 }.

Then for any φ ∈ L p(Ω) with 1 ≤ p ≤ ∞, there is a solution u of the equation

¯

∂u = φ such that u ∈ L p (Ω) Moreover, if p = ∞, there is no solution u ∈ L q(Ω)

with q > p.

The organization of the paper is as follows: we recall the construction of the Henkin solution via the Henkin kernel in Sec 2 We prove Theorem 1.2in Sec 3

and discuss Example 1.1in Sec 4

2 Henkin Solution

In this section, we recall the construction of the Henkin kernel and Henkin solution

to ¯∂ For complete details, see [10, 18], or for a more modern treatment, see [5] Our construction follows the lines of Khanh [12] and Range [Chap V, Sec 1, 18]

to build a Leray map that focuses on the local behavior of the kernel

Definition 2.1 A C2-valuedC1 function G(ζ, z) = (g1(ζ, z), g2(ζ, z)) is called a Leray map for Ω if g1(ζ, z)(ζ1− z1) + g2(ζ, z)(ζ2− z2)= 0 for every (ζ, z) ∈ bΩ × Ω.

A support function Φ(ζ, z) for Ω is a smooth function deﬁned near bΩ × ¯Ω so that

Φ admits a decomposition

Φ(ζ, z) = 2

2

j=1

Φj (ζ, z)(ζ j − z j ),

where Φj (ζ, z) are smooth near bΩ × ¯Ω, holomorphic in z, and vanish only on the

diagonal{ζ = z}.

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For a convex domain, it is well known that G(ζ, z) = ∂ρ

∂ζ = (∂ζ ∂ρ1, ∂ρ

∂ζ2) is a Leray map [5, Lemma 11.2.6], and Φ deﬁned by the Leray map

Φj (ζ, z) = ∂ρ(ζ) ∂ζ

j , j = 1, 2,

is a support function for Ω

Let φ =2

j=1 φ j d¯ z j be a bounded,C1, ¯∂-closed (0, 1)-form on ¯Ω The solution

u of the ¯ ∂-equation, ¯ ∂u = φ, provided by the Henkin kernel is given by

u = T φ(z) = Hφ(z) + Kφ(z). (2.1) where

Hφ(z) = 1

2π2

ζ∈bΩ

∂ρ(ζ)

∂ζ1 (¯ζ2− ¯z2)− ∂ρ(ζ) ∂ζ2 (¯ζ1− ¯z1)

Φ(ζ, z) |ζ − z|2 φ(ζ) ∧ ω(ζ),

Kφ(z) = 1

4π2

Ω

φ1(ζ)(¯ ζ1− ¯z1)− φ2(ζ)(¯ ζ2− ¯z2)

|ζ − z|4 ω(¯ ζ) ∧ ω(ζ),

(2.2)

where ω(ζ) = dζ1∧ dζ2 See, for example, [9,7] To understand the L p -norm of u,

it suﬃces to investigate the L p mapping properties of integral operators H and K.

As a consequence of the Riesz–Thorin Interpolation Theorem and Theorem1.1,

proving that T is a bounded, linear operator on L1(Ω) suﬃces to establish that T

is a bounded linear operator on L p(Ω), 1≤ p ≤ ∞.

The L1-estimate of |Kφ(z)| is standard and does not require interpolation.

Indeed, since |ζ − z| −3 ∈ L1(Ω) in both ζ and z (separately), L p boundedness

of K, 1 ≤ p ≤ ∞, follows from [8, Theorem 6.18]

The expression for H given in (2.2) uses the fact that φ is deﬁned on bΩ However, after an integration by parts (done below), H has an expression that is

an integral over Ω Once we show that H is bounded on L1

,1(Ω), then the density

of C1

,1( ¯Ω)∩ ker ¯∂ in L1

,1(Ω)∩ ker ¯∂ ﬁnishes the argument (the density argument

follows by using the argument of [5, Lemma 4.3.2], keeping in mind that a convex domain is star-shaped and we can ignore the technicalities associated with ¯∂ ∗).

Alternatively, Krantz [15, p.257] presents an argument which also discusses how to

pass from estimates on smooth forms on the boundary to merely L p forms in Ω.

For the boundedness of H, we ﬁrst begin the analysis of Hφ(z) by using Stokes’ Theorem Using the assumption that φ is ¯ ∂-closed, we observe

Hφ(z) = 1

2π2

Ω

¯

∂ ζ

Φ1(ζ, z)(¯ ζ2− ¯z2)− Φ2(ζ, z)(¯ ζ1− ¯z1)

(Φ(ζ, z) − ρ(ζ))(|ζ − z|2+ ρ(ζ)ρ(z)) ψ(ρ(ζ))

∧ φ(ζ) ∧ ω(ζ),

where ψ ∈ C ∞(R) is a cutoﬀ function so that 0 ≤ ψ ≤ 1 and

ψ(t) =

1 for t ≥ −δ/8,

0 for t ≤ −δ/4.

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We abuse notation slightly and let H(ζ, z) be the integral kernel of H As a

conse-quence of Tonelli’s Theorem, it suﬃces to prove that

(ζ,z)∈Ω×Ω |H(ζ, z)φ(ζ)|dV (ζ, z) φ L1(Ω)< ∞. (2.3)

Since H(ζ, z) = 0 when ρ(ζ) ≤ −δ/4, it suﬃces to prove

(ζ,z)∈{ζ∈Ω:ρ(ζ)>−δ/4}×Ω |H(ζ, z)φ(ζ)| dV (ζ, z) φ L1(Ω).

Direct calculation shows that

|H(ζ, z)| ≤

¯∂ ζ

Φ1(ζ, z)(¯ ζ2− ¯z2)− Φ2(ζ, z)(¯ ζ1− ¯z1)

(Φ(ζ, z) − ρ(ζ))(|ζ − z|2+ ρ(ζ)ρ(z)) ψ(ρ(ζ))

|Φ(ζ, z) − ρ(ζ)|2 1

(|ζ − z|2+ ρ(ζ)ρ(z))1/2

|Φ(ζ, z) − ρ(ζ)|(|ζ − z|2+ ρ(ζ)ρ(z)) , (2.4)

recognizing that

|ζ − z|

(|ζ − z|2+ ρ(ζ)ρ(z)) ≤ |ζ − z|

|ζ − z|(|ζ − z|2+ ρ(ζ)ρ(z))1/2

(|ζ − z|2+ ρ(ζ)ρ(z))1/2 .

Since ρ is smooth, ρ is Lipschitz, so (ρ(ζ) − ρ(z))2 |ζ − z|2 Therefore, ρ(ζ)2

|ζ − z|2+ ρ(ζ)ρ(z), hence |ζ − z| + |ρ(ζ)| (|ζ − z|2+ ρ(ζ)ρ(z))1/2 Thus

|Φ(ζ, z) − ρ(ζ)| |ζ − z| + |ρ(ζ)| (|ζ − z|2

+ ρ(ζ)ρ(z))1/2 . (2.5)

Combining (2.4) and (2.5), we obtain

|Φ(ζ, z) − ρ(ζ)|2(|ζ − z|2+ ρ(ζ)ρ(z))1/2

≤ |Φ(ζ, z) − ρ(ζ)|1 2|ζ − z| (2.6)

Since bΩ is compact, there exists δ > 0 and points p1, , p N ∈ bΩ so that

bΩ is covered by {B(p j , δ) } N

j=1 After changing coordinates to set p j to 0 with the

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transformation T p j in Sec 1, we may assume the goal is to prove

(ζ,z)∈(Ω pj ∩B(0,δ))×Ω pj |H(T −1

p j (ζ p j ), T −1

p j (z p j ))φ(T −1

p j (ζ p j))| dV (ζ p j , z p j)

φ(T −1

p j (·)) L1(Ωpj)≈ φ L1(Ω), (2.7) where

Ωp j ={ρ p j (z p j ) := ρ(T −1

p j (z p j )) = P p j (z p j ,1 ) + r p j (z p j ) < 0 }

and P p j (z p j ,1 ) = F p j(|z p j ,1 |2) or P (z p j ,1 ) = F (x2p j ,1) as in Sec 1 Since

Φ(T −1

p j (ζ p j ), T −1

p j (z p j)) = Φp j (ζ p j , z p j ),

where, Φp j is the support function of Ωp j, we obtain

|H(T −1

p j (ζ p j ), T −1

p j (z p j))| 1

|Φ p j (ζ p j , z p j)− ρ p j (ζ p j)|2|ζ p j − z p j | . (2.8)

Here and in what follows, we omit the subscript p j and still use φ( ·) for φ(T −1

p j (·)).

3 Proof of Theorem 1.2

We will investigate the complex and real ellipsoid cases separately to show (2.7)

First, however, we recall the following facts for the class of real functions F in the ﬁrst part with the additional assumption that F (0) = 0 See, for example,

[9, Lemma 4]

Lemma 3.1 Let F be a C2 convex function on [0, d] Then

F (p) − F (q) − F (q)(p − q) ≥ 0 (3.1)

for any p, q ∈ [0, d] If, in addition, F (0) = 0 and F is nondecreasing, then

F (p) − F (q) − F (q)(p − q) ≥ F (p − q), (3.2)

for any 0 ≤ q ≤ p ≤ d.

Proof The proof of (3.1) is simple and is omitted here For (3.2), let s := p −q ≥ 0

and g(s) := F (s + q) −F (q)−sF (q) −F (s) Hence, g (s) = F (s + q) −F (q) −F (s)

and g (s) = F (s + q) − F (s) Using the assumption F (t) is nondecreasing, we

have g (s) ≥ 0, thus g (s) is nondecreasing This implies g (s) ≥ g (0) = 0 (since

F (0) = 0) and consequently that g(s) is increasing We thus obtain g(s) ≥ g(0) = 0

(since F (0) = 0) This completes the proof of (3.2)

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3.1. Complex ellipsoid case

In this subsection, Ω is deﬁned by (1.1) Since the argument of F is |ζ1|2, the chain rule shows that ∂

∂ζ1F ( |ζ1|2) = ¯ζ1F (|ζ1|2) Similarly to Khanh [12, (4.1)], the

convexity of r shows that

Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1|2)− F (|ζ1|2)− 2F (|ζ1|2)Re{¯ζ1(z1− ζ1)}

=−ρ(z) + F (|ζ1|2)|z1− ζ1|2+ (F ( |z1|2)− F (|ζ1|2)

− F (|ζ1|2)(|z1|2− |ζ1|2)). (3.3)

The analysis splits into two cases: (i) F (0)= 0 and (ii) F (0) = 0 In the ﬁrst case,

the hypotheses on F guarantee the existence of a δ > 0 such that F (|ζ1|2) > 0 for

any|ζ1| < δ Hence,

Re{Φ(ζ, z)} − ρ(ζ) −ρ(z) + |z1− ζ1|2

and

(|ρ(z)|2+|ImΦ(ζ, z)|2+|ζ1− z1|4)|ζ1− z1| .

The estimate in this case is the estimate for the case of a strongly pseudoconvex domain, and the result is classical and well-known Thus, we may assume that

F (0) = 0.

Lemma 3.2 Let F be defined as in Sec.1 with the additional assumption F (0) =

0 Then, for any |z1|, |ζ1| ≤ d

|H(ζ, z)|











1 (|ρ(z) + iImΦ(ζ, z)|2+ F2(|z1− ζ1|2))|z1− ζ1| if |ζ1| ≥ |z1− ζ1|,

1 (|ρ(z) + iImΦ(ζ, z)|2+ F2(12|z1|2))|z1| if |ζ1| ≤ |z1− ζ1|.

(3.4)

Proof Applying Lemma3.1to (3.3), we obtain

Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) +







F (|ζ1|2)|z1− ζ1|2 if 0 < |z1|, |ζ1| < d,

F ( |z1|2− |ζ1|2) if|ζ1| ≤ |z1| ≤ d. (3.5)

We now compare the relative sizes of|ζ1| and |z1− ζ1|.

Case 1.|ζ1| ≥ |z1− ζ1| Combining the ﬁrst inequality from (3.5) with (ii) from

p 2, we obtain

Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1− ζ1|2).

The ﬁrst line of (3.4) follows by this inequality and (2.6)

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Case 2.|ζ1| ≤ |z1− ζ1| In this case, the estimate depends on the relative sizes of

|ζ1| and √1

2|z1| If |ζ1| ≥ √1

2|z1|, then the argument from Case 1 proves that

Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1− ζ1|2)|z1− ζ1|2≥ −ρ(z) + F

1

2|z1|2

,

and we obtain the second estimate in (3.4) Otherwise, |ζ1| ≤ √1

2|z1|, and this

implies both|z1| ≥ |ζ1| and |z1− ζ1| ≥ (1 − √1

2)|z1| By the second case of (3.5),

we observe that

Re{Φ(ζ, z)} − ρ(ζ) ≥ −ρ(z) + F (|z1|2− |ζ1|2)≥ −ρ(z) + F

1

2|z1|2

and

(|Re{Φ(ζ, z)} − ρ(ζ)|2+|ImΦ(ζ, z)|2)|ζ1− z1|

|ρ(z) + iImΦ(ζ, z)|2+ F2

1

2|z1|2

This completes the proof

Proof of Theorem 1.2 (1) From (2.7), the only remaining estimate is the integral

over (ζ, z) ∈ (Ω∩B(0, δ))×Ω We now concentrate on the integral near 0 By Lemma

3.2, we have

(ζ,z)∈(Ω∩B(0,δ))×Ω |H(ζ, z)φ(ζ)| dV (ζ, z)

=

(ζ,z)∈(Ω∩B(0,δ))×Ω and |ζ1|≥|z1−ζ1| · · ·

+

(ζ,z)∈(Ω∩B(0,δ))×Ω,|ζ1|≤|z1−ζ1| and |z1|≤2δ · · ·

+

(ζ,z)∈(Ω∩B(0,δ))×Ω,|ζ1|≤|z1−ζ1| and |z1|≥2δ · · ·

where

(I) :=

(ζ,z)∈(B(0,2δ)∩Ω)2

|φ(ζ)|dV (ζ, z)

(|ρ(z) + iImΦ(ζ, z)|2+ F2(|z1− ζ1|2))|z1− ζ1| ,

(II) :=

(ζ,z)∈(B(0,2δ)∩Ω)2

|φ(ζ)|dV (ζ, z)

(|ρ(z) + iImΦ(ζ, z)|2+ F2(12|z1|2))|z1| ,

(III) :=

(ζ,z)∈(B(0,δ)∩Ω)×Ω and |z1|≥2δ

|φ(ζ)|dV (ζ, z)

(|ρ(z) + iImΦ(ζ, z)|2+ F2(12|z1|2))|z1| .

(3.8)

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It is easy to see that (III) (F (2δ2)δ) −1 φ L1(Ω) For the integral (I), we make the

change of variables (α, w) = (α1, α2, w1, w2) = (ζ1, ζ2, z1− ζ1, ρ(z) + iImΦ(ζ, z)).

The Jacobian of this transformation is the matrix

J =







∂(Rez1 )

∂ImΦ(ζ,z)

∂(Reζ1 )

∂ImΦ(ζ,z)

∂(Imζ1 )

∂ImΦ(ζ,z)

∂(Reζ2 )

∂ImΦ(ζ,z)

∂(Imζ2 )

∂ImΦ(ζ,z)

∂(Rez1 )

∂ρ(z)

∂(Imz1 )

∂ρ(z)

∂(Rez2 )

∂ρ(z)

∂(Imz2 )

∂ImΦ(ζ,z)

∂(Imz1 )

∂ImΦ(ζ,z)

∂(Rez2 )

∂ImΦ(ζ,z)

∂(Imz2 )







To justify this coordinate change, we write z j = x j + iy j and compute

det(J ) = ∂ Im(Φ(ζ, z))

∂y2

∂ρ(z)

∂x2 − ∂ Im(Φ(ζ, z))

∂x2

∂ρ(z)

∂y2

.

By a possible rotation and dilation of Ω, we can assume that∇ρ(0) = (0, 0, 0, −1).

A direct calculation then establishes that if δ is chosen suﬃciently small (so that

∂ρ(z)

∂y2 dominates the other partials of ρ and |ζ − z| ≤ 4δ is small), then det(J) = 0.

Since Φ is smooth, we can assume that there exists δ > 0 that depends on Ω, δ,

and ρ so that

(I)

(α,w)∈(Ω∩B(0,δ ))×B(0,δ )

|φ(α)|

(|w2|2+ F2(|w1|2))|w1| dV (α, w)

φ L1(Ω)

δ

0

δ

0

r1r2

(r2+ F2(r2))r1

dr2dr1

φ L1(Ω)

δ

0

log F (r21) dr1 < ∞.

That the integral is ﬁnite follows by the hypotheses on φ and F

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transformation T p j in Sec 1, we may... example for studying necessary and suﬃcient conditions for local regularity of ¯∂ on domains of< /i>

inﬁnite type In fact, the canonical solution of ¯∂ associated to this tube domain... recently, there has been work on supnorm estimates for the Cauchy– Riemann equations on inﬁnite type domains in C2 Fornæss et al provided the< /i>

ﬁrst examples in [9] and Khanh

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