Hölder Type Estimates for the <span class="mathjax-formula">$\overline{\partial }$</span>-equation in Strongly Pseudoconvex Domains

HoÈlder Type Estimates for the @-Equation in Strongly Pseudoconvex Domains.

HEUNGJUAHN(*) - HONGRAECHO(**) - JONG-DOPARK(***)

ABSTRACT- Inthis paper we generalize the HoÈlder space with a majorant function, define its order, and prove the existence and regularity for the solutions of the Cauchy-Riemann equation in the generalized HoÈlder space over a bounded strongly pseudoconvex domain.

1. Introduction and regular majorant.

IfD is a bounded domain in Cⁿ, the HoÈlder space of order a, La(D) (0<a<1), is defined as the set of all functionsgonDwhich satisfy for a constantCˆCg>0 the condition

jg(z) g(z)j Cjz zj^a; z;z2D:

We first generalize this HoÈlder space following Dyakonov [Dya97] (also see PavlovicÂ's book [Pav04]). For this purpose we introduce the notion of a regular majorant. Let v be a continuous increasing function on [0;1). We assume v(0)ˆ0, and suppose that v(t)=t is non-increasing

(*) Indirizzo dell'A.: Nims 3F Tower Koreana, 628 Daeduk-Boulevard, Yuseong- gu, Daejeon305-340, South Korea.

E-mail: heungju@nims.re.kr

(**) Indirizzo dell'A.: Department of Mathematics, Pusan National University, Geumjeong-gu, Busan 609-735 Korea.

E-mail: chohr@pusan.ac.kr

(***) Indirizzo dell'A.: Department of Mathematics, Pohang University of Science and Technology San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk 790- 784, Korea.

E-mail: jongdopark@gmail.com

Partially supported by Korea Research Foundation Grant 2005-070-C00007.

2000 Mathematics Subject Classification: Primary 32A26, 32W05.

and satisfies the inequality Z^d

0

v(t) t dt‡d

Z¹

d

v(t)

t² dtCv(d); for any 0<d<1;

(1:1)

for a suitable constant CˆC(v). Such a function v is called a regular majorant. Givena regular majorantv, the HoÈlder type space,L(v;D) is defined as the family of all functionsg onDsuch that

jg(z) g(z)j Cv(jz zj); z;z2D:

(1:2)

The normkgk_vofg2L(v;D) is givenbyCg‡ kgk₁, whereCg0 is the smallest constant satisfying (1.2) andkgk₁is theL¹norm inD. Note that with this norm L(v;D) is a Banach space andL(v;D)L¹(D) (see the chapter 10 of PavlovicÂ's book [Pav04]). We denote byL_q(v;D) the set of the differential forms of type (0;q) whose coefficients are inL(v;D). We define theorderof a regular majorant as follows:

DEFINITION 1.1. We say that a regular majorant v has order a (0<a<1) if there exist anaand a positive real numbert₀such that

aˆsup g:v(t)

t^g is increasing8t; 0<t<t0

ˆinf g:v(t)

t^g is decreasing8t; 0<t<t₀

:

If a regular majorantvhasordera, thenwe letvˆva and callL(va;D) the HoÈlder type space of ordera. By definition of the orderof a regular majorant, it is uniquely determined, if it exists. Now we state our main result of this paper.

THEOREM1.2. Let DCⁿ (n2)be a strongly pseudoconvex do- main with C⁴-boundary and0<a<1=2. If a regular majorantv_a has orderaand f 2L_q(v_a;D)with@f ˆ0 (1qn),then there is a solution u2L(q 1)(t¹⁼²va;D)of @uˆf such that for some constant CˆC(va)

kuk_t1=2vaCkfk_va: (1:3)

The above inequality (1.3) generalizes the estimate by Henkin-Roma- nov [RH71] and Lieb-Range [LR80]

kuk_ta‡1=2Ckfk_ta:

For the proof of the HoÈlder type estimate (1.3), we need a variant of the Hardy-Littlewood Lemma [HL84].

LEMMA1.3. Let DRⁿbe a bounded domain with C¹-boundary. If g is a C¹(D)-function andv_gis a regular majorant of orderg,0<g<1such that for some constant cgdepending on g,

jdg(x)j cg vg(jr(x)j)

jr(x)j ; x2D;

then we have

jg(x) g(y)j cg vg(jx yj):

As convention we use the notationA9BorA0Bif there are constants c1;c2, independent of the quantities under consideration, satisfying Ac₁BandAc₂B, respectively.

Before proving our theorem, we discuss some properties of a regular majorant and some examples.

EXAMPLE 1.4. (i) The most typical example is a function v(t)ˆt^a (0<a<1). Clearly,vis a regular majorant and has ordera.

(ii) A non-trivial example is the function, v(t)ˆt^ajlogtj^b on[0;t0] ex- tended continuously for t>t0 to be a regular majorant. Here 0<a<1, 1<b<1andt₀must be chosensufficiently small so that the function vshould be a regular majorant (t₀depends ona;b). Since lim

t&0t^ejlogtj^bˆ0 for anye>0, it follows thatv(t)ˆt^ajlogtj^bhas orderafor any choice ofb.

(iii) Define the function m(t)ˆ1=jlogtj^b, b>0 for 0<t<t0 and m(0)ˆ0. Then m(t) is continuous and increasing near 0, but it is not a regular majorant.

We end this section by describing useful properties of a regular ma- jorant.

REMARK1.5. (i) Ifv;mare two regular majorants and have ordersa;b respectively with (0<a<b<1), thenletting v_a;m_b, there existt₀>0 and c such that m_b(t)cv_a(t), 0tt₀. Hence we have the inclusion L(mb;D)L(va;D). Note that if two regular majorants, v;m have the same ordera, then generally there is no inclusion relation betweenL(v;D) andL(m;D).

(ii) Inour Theorem 1.2, for a general regular majorant of order 1=2, the

estimate kuk_v1=29kfk₁ does not hold. In fact, the celebrated Henkin's theorem [RH71] holds only for the special regular majorantv₁₌₂(t)ˆ jtj¹⁼² and this number 1=2 is the sharp bound [Ran86]. But there is a regular majorantm₁₌₂(t)ˆ jtj¹⁼²jlogtjnear the origin of order 1/2, which is strictly bigger thanjtj¹⁼².

REMARK1.6. Letvbe a regular majorant of ordera(0<a<1=2), say vˆva. Thent¹⁼²va is also a regular majorant of order (a‡1=2). Infact, t¹⁼²v_a is increasing and (t¹⁼²v_a)=t is non-increasing, since v_a=t^g,g>a is decreasing. Here we use the fact thatv_a has ordera. It remains to show thatt¹⁼²v_aalso satisfies (1.1). Since (t¹⁼²v_a)=tis non-increasing, we have for anyd, (0<d<1),

Z^d

0

s¹⁼²va(s)

s ds9d¹⁼² Z^d

0

va(s)

s ds9d¹⁼²v_a(d):

(1:4)

Onthe other hand, for a given0<a<1=2, we canchoose a sufficiently smallesuch thata<1=2 e. It follows from the order ofv_athatv_a=t^{(1=2 e)} is decreasing. Hence we obtain

d Z¹

d

s¹⁼²va(s) s² dsˆd

Z¹

d

va(s) s^{1=2 e}

1 s^1‡e ds

9dv_a(d) d^{1=2 e}

Z¹

d

1

s^1‡e ds9d¹⁼²v_a(d):

(1:5)

By (1.4) and (1.5),t¹⁼²v_ais a regular majorant of order (a‡1=2).

Acknowledgments. The authors are grateful to the referee for several valuable suggestions.

2. Henkin's solution operator of the@-equation.

In this section, we introduce the Henkin's solution operator [HL84] of

the@-equation and prove the integral estimates for the solution operator in

a strongly pseudoconvex domain inCⁿ. LetDbe defined by a functionr, i.e.,Dˆ fz2Cⁿ:r(z)<0g, wherer2C⁴ andrr6ˆ0 onbD.

To construct the integral formula for solutions of the @-equationina strongly pseudoconvex domain, we need a support function (see [HL84]).For

the global support function, we follow Fornaess construction [For76]. He showed that there exist a neighborhood U of D and a function f(;)2C³(UU) such that for allz2U,f(z;) is holomorphic inU and f(z;z)ˆ hF;z zi, where we define FˆF(z;z)ˆ(f₁(z;z);. . .;f_n(z;z)) andhF;z zi ˆPⁿ

jˆ1f_j(z;z)(z_j z_j). In[For76], Fornaess also showed that f_j 2C³(UU) is holomorphic in z and there is a constant c such that for all z2D and z2D we have

2Ref(z;z)r(z) r(z)‡cjz zj² (2:6)

andd_zf(z;z)j_zˆz ˆ@r(z). Suppose thatf 2L_q(v_a;D) (1qn) an d@f ˆ0.

Then f is uniformly continuous in D. Using the above global support functionf, we define Henkin kernelH(z;z) as follows:

H(z;z)ˆ 1 (2pi)ⁿ

hz z;dzi

jz zj² ^ hF;dzi

hF;z zi^ X

k‡`ˆn 2

hz z;dzi jz zj²

!_k

^ h@z;zF;dzi jz zj²

!_`

; where@_z;zFˆ@_zFanddzˆ(dz₁;. . .;dz_n). Note thatFis holomorphic inz.

We also define the Bochner-Martinelli kernel:

K(z;z)ˆ 1 (2pi)ⁿ

hz z;dzi

jz zj² ^ hdz dz;dzi jz zj²

!_{n 1} :

For the construction of the above kernels, see [Ran86] or [CS01]. We have the Henkin's solution operatorSf ˆKf Hf of the@-equation, where

Hf(z)ˆ Z

z2bD

f(z)^H(z;z); Kf(z)ˆ Z

z2D

f(z)^K(z;z):

We remark that the fact that the support functionf(z;z) is holomorphic inz is very crucial inthe constructionof the solutionoperator Sf of the @- equation.

To prove the HoÈlder type estimate (1.3) of the mainTheorem 1.2, we use Lemma 1.3. Hence, we have to estimate the differential of the Henkin solutionoperator, d_zSf. Using the fact that jz zj²9jf(z;z)j for (z;z)2bDD, straightforward computations give the kernel estimate (for the details, see [Ran86])

jd_zH(z;z)j9 1

jf(z;z)j²jz zj^{2n 3}; (z;z)2bDD:

REMARK 2.1. Without loss of generality, we assume that the differ- ential of Henkin kernel,dzH(z;z) has the following form:

dzH(z;z)ˆ A(z;z) (f(z;z))²jz zj^{2n 2}; (2:7)

where A(;z) belongs to C¹(D) and satisfies jA(z;z)j9jz zj. Actually, d_zH(z;z) contains more terms whose singularity order is lower than that of (2.7) and so we can ignore other terms (refer to § 3. of chapter 4 in [Ran86]).

Generally, the Bochner-Martinelli integral,Kf, has a good regularity, soKis a bounded operator fromL¹-forms toL_a-forms for any 0<a<1.

This kind of regularity still holds for a regular majorant of order a (0<a<1). Hence, we only prove the estimate for the differential of Henkin kernel,dzHf, which is the mainpart of this paper.

PROPOSITION2.2. For anyawith0<a<1=2,there exists a constant C_a>0such that

jdzHf(z)j Cakfk_vav_a(jr(z)j)

jr(z)j¹⁼² for z2D:

(2:8)

PROOF. Since the singularities of the Henkin kernel are located in the diagonalbDbD, to show the inequality (2.8), it suffices to estimate the integral of (2.8) near boundary points. Fix a pointz2Dwhich is sufficiently close to the boundary ofDand choose a ballB(z;r) withB(z;r)\bD6ˆ ;, inwhich we have a C¹ coordinates system (t₁;. . .;t_2n)ˆtˆt(z;z) such that t₁ˆ r(z);t₂ˆImf(z;z);t(z;z)ˆ( r(z);. . .;0), and jt(z;z)j<1 for z2B(z;r). (For the detail, see [HL84].) Moreover, this coordinate system tsatisfies

jtj9jz zj9jtj; z2B(z;r)\bD:

Also, note that the new coordinate system satisfies t(z;z)ˆ(0;t⁰) for z2B(z;r)\bD, wheret⁰ˆ(t₂;. . .;t_2n). By Remark 2.1, we have to show that

I(z)ˆ

Z

bD\B(z;r)

f(z)x(z)A(z;z)

(f(z;z))²jz zj^{2n 2} dV(z)

9kfk_vava(d(z)) d(z)¹⁼² ; where x is a compactly supported cut-off function in B(z;r). For this kind of estimate of HoÈlder type, we choose z⁰2B(z;r)\bD satisfying t(z⁰;z)ˆ(0;0;t₃;. . .;t_2n). This gives the obvious estimate, I(z)I₁(z)‡

‡I2(z), where

I₁(z)ˆ Z

bD\B(z;r)

(f(z) f(z⁰))x(z)A(z;z) (f(z;z))²jz zj^{2n 2} dV(z)

; I2(z)ˆ

Z

bD\B(z;r)

f(z⁰)x(z)A(z;z)

(f(z;z))²jz zj^{2n 2} dV(z) :

It follows from the definition of k k_va and the inequality jA(z;z)j9 9jz zj, that

I₁(z) kfk_va Z

bD\B(z;r)

va(jz z⁰j)

jf(z;z)j²jz zj^{2n 3}dV(z):

(2:9)

To estimate the integral of the right hand side of (2.9), we use the co- ordinate systemt, the inequality (2.6), and introduce polar coordinates in t⁰⁰ˆ(t₃;. . .;t_2n)2R^{2n 2}, and also setrˆ jt⁰⁰j. Thenwe have

I1(z)9kfk_va Z

jt⁰j<1

va(jt2j)

(jt2j ‡ jt⁰j²‡ jr(z)j)²jt⁰j^{2n 3} dV(t⁰)

9kfk_va Z

jt2j<1

v_a(jt₂j) Z¹

0

r^{2n 3}dr (jt2j ‡r²‡ jr(z)j)²r^{2n 3} 2

4

3 5dt₂

9kfk_va Z¹

0

v_a(t₂)

(t2‡ jr(z)j)³⁼² dt2:

We may assume that 0<jr(z)j<1, sincez2Dis close to the boundary.

We decompose the integral as follows:

Z¹

0

v_a(t₂)

(t2‡ jr(z)j)³⁼² dt2ˆ Z

jr(z)j

0

v_a(t₂)

(t2‡ jr(z)j)³⁼² dt2‡ Z¹

jr(z)j

v_a(t₂)

(t2‡ jr(z)j)³⁼² dt2: Since v_a is a regular majorant, by the first term of the left hand side of (1.1), we have

Z^jr(z)j

0

v_a(t₂)

(t₂‡ jr(z)j)³⁼²dt₂9 1 jr(z)j¹⁼²

Z^jr(z)j

0

v_a(t₂)

t2 dt₂9v_a(jr(z)j) jr(z)j¹⁼² :

Similarly, sinces¹⁼²v_a(s) is also a regular majorant, by the second term of

the left hand side of (1.1), it follows that Z¹

jr(z)j

va(t2)

(t2‡ jr(z)j)³⁼²dt2ˆ Z¹

jr(z)j

t¹⁼²₂ v_a(t₂) t²₂

t³⁼²₂

(t2‡ jr(z)j)³⁼² dt2

9 Z¹

jr(z)j

t¹⁼²₂ va(t2) t²₂ dt₂

9jr(z)j¹⁼²v_a(jr(z)j)

jr(z)j ˆv_a(jr(z)j) jr(z)j¹⁼² : These inequalities implyI₁(z)9kfk_vav_a(jr(z)j)=jr(z)j¹⁼².

For I₂(z), we need a somewhat different method. The integration by parts allows one to lower the singularity order of the Henkin kernel. This kind of method was used in [Ran92].

We see that

1

f²ˆ @f

@t2 1 @

@t2

1 f : Therefore, by integration by parts, we have

I2(z)

Z

jt⁰j1

@f

@t2 1 @

@t2

1 f

f(0;0;t⁰⁰)x(t⁰)A(t⁰;z) jt⁰j^{2n 2} dt⁰

(2:10)

ˆ

Z

jt⁰j1

f(0;0;t⁰⁰)1 f

@

@t₂

@f

@t₂

1x(t⁰)A(t⁰;z) jt⁰j^{2n 2}

" # dt⁰

ˆ

Z

jt⁰j1

f(0;0;t⁰⁰)1 f

@f

@t₂

2B(t⁰;z)dt⁰ ;

where

B(t⁰;z)ˆ @²f

@t²₂

x(t⁰)A(t⁰;z) jt⁰j^{2n 2} ‡@f

@t₂

@

@t₂

x(t⁰)A(t⁰;z) jt⁰j^{2n 2}

! :

Inthe second equality of (2.10), we use the fact thatf(0;0;t⁰⁰) does not de- pend ont₂. Sincet₂ˆImf, we havej@f=@t₂j 1:Therefore, we have

I2(z)9kfk₁ Z

jtj⁰j1

dt⁰ jfkt⁰j^{2n 2}:

For the moment, we assume that for anye>0, J(z)ˆ

Z

jt⁰j1

dt⁰

jfkt⁰j^{2n 2}9jr(z)j ^e; (2:11)

which will be proved later as an independent lemma. Since (2.11) holds for arbitrarye>0, one can choosee>0 so that 0<1=2 e<a. Moreover,v_a, 0<a<1=2, is a regular majorant and so v_a(t)=t^{1=2 e} is increasing, or equivalently,jr(z)j ^e9va(jr(z)j)=jr(z)j¹⁼². It follows that

I2(z)9kfk₁J(z)9kfk_vava(jr(z)j) jr(z)j¹⁼² :

These two estimates forI1(z) an dI2(z) complete the proof. p We end this section with the proof of (2.11).

LEMMA2.3. For anye>0,there exists C_e>0such that J(z)Cejr(z)j ^e:

PROOF. We have J(z)9

Z

jt⁰j<1

dt⁰

(jt2j ‡ jr(z)j ‡ jt⁰j²)jt⁰j^{2n 2} 9

Z

j(t2;t3;t4)j<1

dt2dt3dt4

(jt2j ‡ jr(z)j)(t²₂‡t²₃‡t²₄):

Again, using polar coordinates in (t3;t4), sayxˆ j(t3;t4)j, one obtains

J(z)9 Z

jt2j<1

1 jt₂j ‡ jr(z)j

Z¹

0

x dx t²₂‡x² 0

@

1 Adt₂

9 Z¹

0

jlogt₂j (t₂‡ jr(z)j)dt₂

Ce

Z¹

0

t₂^e

(t2‡ jr(z)j)dt2:

By the change of variablesˆt2=jr(z)j, we have J(z)9

Z¹

0

jt2j ^e (jt₂j ‡ jr(z)j)dt2

9jr(z)j ^e Z¹

0

ds

(1‡s)s^eC_ejr(z)j ^e: 3. Proof of Theorem 1.2. p

Inthis section, we complete the proof of our mainTheorem 1.2 using Proposition2.2 and Lemma 1.3.

The inequality (2.8) in Proposition 2.2 implies that jdHf(z)j cakfk₁jr(z)j¹⁼²v_a(jr(z)j) jr(z)j :

Therefore by Lemma 1.3 and regularities of the operatorHf inthe HoÈlder type spaces we canprove the inequality (1.3) of Theorem 1.2.

Finally, we include a brief sketch of the proof of Lemma 1.3.

PROOF. Because D is compact, by the local coordinate change argument, it suffices to show the following in the special domain D(k)ˆ f(x₁;x⁰)2Rⁿ :0<x₁<k; jx⁰j<kg: if

jdg(x)j c_gv_g(x₁) x1

(3:12)

forx;y2D(k=2) withjx yj k=2, thenwe have jg(x) g(y)j ccgvg(jx yj):

(3:13)

To show this, fix two points x;y2D(k=2) with jx yj k=2 and let dˆ jx yj. Here we may assume thatk1=2 and by symmetry we may also supposex₁y₁.

First it follows from (3.12) that jg(x1;x⁰) g(x1‡d;x⁰)j

Z

x1‡d

x1

@g

@x1(t;x⁰)dt (3:14)

c_g Z

x1‡d

x1

v_g(t)

t dtcc_g v_g(d)

Infact, if 0<dx1, then Z

x1‡d

x1

v_g(t)

t dtd v_g(x₁)

x1 v_g(d);

sincevg(t)=tis decreasing. If 0<x₁d, then Z

x1‡d

x1

v_g(t) t dt9

Z^d

0

v_g(t)

t dt9vg(d):

(3:15)

Sincev_g(t)=tis decreasing, the first inequality of (3.15) holds and by (1.1) the second inequality of ( 3.15) is also true.

Next, by the Mean Value Theorem and (3.12), sincevg(t)=tis decreas- ing, we have

jg(x₁‡d;x⁰) g(y₁‡d;y⁰)j c_gdvg(a1)

a₁ c_gv_g(d) (3:16)

for somea₁in the line segment betweenx₁‡dandy₁‡d. Since jg…x† g…y†j jg…x₁;x⁰† g…x₁‡d;x⁰†j

‡ jg…x₁‡d;x⁰† g…y₁‡d;y⁰†j ‡ jg…y₁‡d;y⁰† g…y₁;y⁰†j;

(3.13) follows from the estimates (3.14) and (3.16). p

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Manoscritto pervenuto in redazione il 10 aprile 2007.