L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Pham Hoang HIEP

Hölder continuity of solutions to the Monge-Ampère equations on compact Kähler manifolds

Tome 60, n^o5 (2010), p. 1857-1869.

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HÖLDER CONTINUITY OF SOLUTIONS TO THE MONGE-AMPÈRE EQUATIONS ON COMPACT

KÄHLER MANIFOLDS

by Pham Hoang HIEP

Abstract. — We study Hölder continuity of solutions to the Monge-Ampère equations on compact Kähler manifolds. T. C. Dinh, V.A. Nguyen and N. Sibony have shown that the measureωⁿ_uis moderate ifuis Hölder continuous. We prove a theorem which is a partial converse to this result.

Résumé. — Nous étudions la continuité de Hölder des solutions des équations de Monge-Ampère sur des variétés Kählériennes compactes. T. C. Dinh, V.A. Nguyen et N. Sibony ont prouvé queωⁿ_uest modéré siuest Hölder-continue. Nous démon- trons dans quelques cas la réciproque de ce résultat.

1. Introduction

LetX be a compactn-dimensional Kähler manifold equipped with a fun- damental formω satisfyingR

Xωⁿ = 1. An upper semicontinuous function ϕ:X →[−∞,+∞)is calledω-plurisubharmonic (ω-psh) ifϕ∈L¹(X)and ωϕ:=ω+dd^cϕ>0. ByPSH(X, ω)(resp.PSH⁻(X, ω)) we denote the set of ω-psh (resp. negativeω-psh) functions onX. The complex Monge-Ampère equationωⁿ_u =f ωⁿ was solved for smooth positive f in the fundamental work of S. T. Yau (see [31]). Later S. Kołodziej showed that there exists a continuous solution iff ∈L^p(ωⁿ),f >0,p >1(see [24]). Recently in [27]

he proved that this solution is Hölder continuous in this case (see also [18]

for the case X =CPⁿ). In Corollary 1.2 in [16] the authors have shown that the measureω_uⁿ is moderate if uis Hölder continuous. The main re- sult is the following theorem which give a partial answer to the converse problem:

Keywords: Hölder continuity, complex Monge-Ampère operator, ω-plurisubharmonic functions, compact Kähler manifolds.

Math. classification:32W20, 32Q15.

Theorem A. — Let µ be a non-negative Radon measure on X such that

µ(B(z, r))6Ar^2n−2+α,

for all B(z, r) ⊂X (A, α >0 are constants). Then for every f ∈L^p(dµ) withp >1,R

Xf dµ= 1, there exists a Hölder continuousω-psh functionu such thatωⁿ_u =f dµ.

The following results are simple applications of Theorem A:

Corollary B. — Letϕ∈PSH(X, ω)be a Hölder continuous function.

Then for everyf ∈L^p(ω_ϕ∧ωⁿ⁻¹)with p >1, R

Xf ω_ϕ∧ωⁿ⁻¹ = 1, there exists a Hölder continuousω-psh functionusuch thatω_uⁿ=f ωϕ∧ωⁿ⁻¹.

Corollary C. — Let S be a C¹ smooth real hypersurface in X and VS be the volume measure onS. Then for every f ∈L^p(dVS)with p >1, R

Xf dVS = 1, there exists a Hölder continuous ω-psh functionusuch that ωⁿ_u =f dVS.

Acknowledgments. — The author is grateful to Slawomir Dinew and Nguyen Quang Dieu for valuable comments. The author is also indebted to the referee for his useful comments that helped to improve the paper.

2. Preliminaries

First we recall some elements of pluripotential theory that will be used throughout the paper. Details can be found in [2]–[3], [5]–[6], [4], [7], [9]–[8], [13]–[15], [19]–[20], [21], [23]–[27], [28], [29]–[30], [32]–[33].

2.1.In [24] Kołodziej introduced the capacityCX onX by C_X(E) := supnZ

E

ωⁿ_ϕ:ϕ∈PSH(X, ω), −16ϕ60o for all Borel setsE⊂X.

2.2.In [19] Guedj and Zeriahi introduced the Alexander capacityTX on X by

T_X(E) =e^{−supXVE,X∗}

for all Borel setsE ⊂X. HereV_E,X^∗ is the global extremalω-psh function for E defined as the smallest upper semicontinuous majorant ofVE,X i.e,

V_E,X(z) = supn

ϕ(z) :ϕ∈PSH(X, ω), ϕ60 onEo .

2.3.The following definition was introduced in [18]: A probability mea- sureµonX is said to satisfy the condition H(α, A)(α, A >0) if

µ(K)6ACX(K)^1+α, for any Borel subsetK ofX.

A probability measure µon X is said to satisfy the condition H(∞) if for anyα >0 there existA(α)>0dependent onαsuch that

µ(K)6A(α)C_X(K)^1+α, for any Borel subsetK ofX.

2.4.The following definition was introduced in [17]: A measureµis said to be moderate if for any open set U ⊂ X, any compact set K ⊂⊂ U and any compact familyF of plurisubharmonic functions on U, there are constantsα >0 such that

supnZ

K

e^−αϕdµ:ϕ∈ Fo

<+∞.

2.5.The following class of ω-psh functions was investigated by Guedj and Zeriahi in [20]:

E(X, ω) =n

ϕ∈PSH(X, ω) : lim

j→∞

Z

{ϕ>−j}

ω_max(ϕ,−j)ⁿ = Z

X

ωⁿ = 1o . Let us also define

E⁻(X, ω) =E(X, ω)∩PSH⁻(X, ω).

We refer to [20] for the properties of the classE(X, ω).

2.6.S is called a C¹ smooth real hypersurface in X if for all z ∈ X there exists a neighborhoodU ofzandχ∈C¹(U)such thatS∩U ={z∈ U:χ(z) = 0} andDχ(z)6= 0for allz∈S∩U.

Next we state a well-known result needed for our work.

2.7. Proposition. — Let µ be a non-negative Radon measure on X such that µ(B(z, r)) 6 Ar^2n−2+α for all B(z, r) ⊂ X (A, α > 0 are constants). Thenµ∈ H(∞).

Proof. — By Theorem 7.2 in [33] and Proposition 7.1 in [19] we can find , C >0such that

µ(K)6Ah^2n−2+α(K)6 AC

α T_X(K)^α6ACe α e

− ^α

CX(K)1 n,

for all Borel subsets K of X, where h^2n−2+α is the Hausdorff content of dimension2n−2 +α. This implies that µ∈ H(∞).

3. Stability of the solutions

The stability estimate of solutions to the Monge-Ampère equations on compact Kähler manifolds was obtained by Kołodziej ([24]). Recently, in [12] S. Dinew and Z. Zhang proved a stronger version of this estimate. We will show a generalization of the stability theorem by S. Kołodziej. As a first step we have the following proposition. This proof follows ideas of the proof of Theorem 2.5 in [11]. We include a proof for the reader’s convenience.

3.1. Proposition. — Letϕ, ψ∈ E⁻(X, ω)be such thatωⁿ_ϕ∈ H(α, A).

Then there exist constantst∈RandC(α, A)>0such that Z

{|ϕ−ψ−t|>a}

(ω_ϕⁿ+ωⁿ_ψ)6C(α, A)aⁿ⁺¹,

herea= [R

Xkωⁿ_ϕ−ωⁿ_ψk]

1 2n+3+n+1

1+α. Proof. — Since R

{|ϕ−ψ−t|>a}(ω_ϕⁿ+ω_ψⁿ) 6 2, it suffices to consider the case whenais small. Set

= 1 2infnZ

{|ϕ−ψ−t|>a}

ω_ϕⁿ:t∈Ro Hence

Z

{|ϕ−ψ−t|6a}

ω_ϕⁿ61−2 for allt∈R. Set

t0= supn t∈R:

Z

{ϕ<ψ+t+a}

ωⁿ_ϕ61−o Replacingψbyψ+t₀we can assume thatt₀= 0. ThenR

{ϕ<ψ+a}ω_ϕⁿ61−

andR

{ϕ6ψ+a}ωⁿ_ϕ>1−. Hence Z

{ψ<ϕ+a}

ωⁿ_ϕ= 1− Z

{ϕ+a6ψ}

ω_ϕⁿ

= 1− Z

{ϕ6ψ+a}

ω_ϕⁿ+ Z

{ψ−a<ϕ6ψ+a}

ω_ϕⁿ61−. SinceR

{|ϕ−ψ|6a}ω_ϕⁿ61we can chooses∈[−a+aⁿ⁺², a−aⁿ⁺²]satisfying Z

{|ϕ−ψ−s|<aⁿ⁺²}

ω_ϕⁿ62aⁿ⁺¹.

Replacing ψ byψ+s we can assume that s = 0. One easily obtains the following inequalities

Z

{ϕ<ψ+aⁿ⁺²}

ωⁿ_ϕ61−, Z

{ψ<ϕ+aⁿ⁺²}

ω_ϕⁿ61−, (1)

Z

{|ϕ−ψ|<aⁿ⁺²}

ωⁿ_ϕ62aⁿ⁺¹.

By [20] we can findρ∈ E(X, ω), such that (2) ωⁿ_ρ = 1

1−1_{ϕ<ψ}ωⁿ_ϕ+c1_{ϕ>ψ}ω_ϕⁿ and sup

X

ρ= 0,

(c > 0 is chosen so that the measure has total mass 1). For simplicity of notation we setβ= ⁿ⁺¹_1+α. Set

U =n

(1−a^n+2+β)ϕ <(1−a^n+2+β)ψ+a^n+2+βρ} ⊂ {ϕ < ψo . From Theorem 2.1 in [15] and (2) we get

(3) ω_ϕⁿ⁻¹∧ω_(1−an+2+β)ψ+a^n+2+βρ>(1−a^n+2+β)ω_ϕⁿ⁻¹∧ω_ψ+ a^n+2+β (1−)ⁿ¹ω_ϕⁿ, onU. From Theorem 2.3 in [15], Lemma 2.6 in [11] and (3) we obtain

(1−a^n+2+β) Z

U

ω_ϕⁿ⁻¹∧ωψ+ a^n+2+β (1−)¹ⁿ

Z

U

ω_ϕⁿ

6 Z

U

ω_(1−an+2+β)ψ+a^n+2+βρ∧ωⁿ⁻¹_ϕ

6 Z

U

ω_(1−an+2+β)ϕ∧ω_ϕⁿ⁻¹

= (1−a^n+2+β) Z

U

ω_ϕⁿ+a^n+2+β Z

U

ω∧ωⁿ⁻¹_ϕ

6(1−a^n+2+β)Z

U

ω_ϕⁿ⁻¹∧ω_ψ+ 2a^2n+3+β

+a^n+2+β Z

U

ω∧ωⁿ⁻¹_ϕ .

Hence

(4) 1

(1−)ⁿ¹ Z

U

ωⁿ_ϕ62aⁿ⁺¹+ Z

U

ω∧ωⁿ⁻¹_ϕ .

From Proposition 3.6 in [19] and (4) we get

1 (1−)ⁿ¹

hZ

{ϕ6ψ−aⁿ⁺²}

ωⁿ_ϕ−C1(α, A)aⁿ⁺¹i (5)

6 1 (1−)ⁿ¹

hZ

{ϕ6ψ−aⁿ⁺²}

ωⁿ_ϕ−A[CX({ρ6− 1

2a^β})]^1+αi 6 1

(1−)ⁿ¹ hZ

{ϕ6ψ−aⁿ⁺²}

ωⁿ_ϕ− Z

{ρ6−_2aβ¹ }

ω_ϕⁿi

6 1 (1−)ⁿ¹

Z

U

ωⁿ_ϕ

62aⁿ⁺¹+ Z

U

ω∧ωⁿ⁻¹_ϕ

62aⁿ⁺¹+ Z

{ϕ<ψ}

ω∧ω_ϕⁿ⁻¹, Similarly toρwe defineϑ∈ E(X, ω), such that

ωⁿ_ϑ = 1

1−1_{ϕ<ψ}ω_ϕⁿ+l1_{ψ>ϕ}ω_ϕⁿ and sup

X

ϑ= 0, (l plays the same role ascabove). Set

V =n

(1−a^n+2+β)ψ <(1−a^n+2+β)ϕ+a^n+2+βϑo

⊂ {ψ < ϕ}.

We get

(6) 1

(1−)ⁿ¹ hZ

{ψ6ϕ−aⁿ⁺²}

ωⁿ_ϕ−C1(α, A)aⁿ⁺¹i

62aⁿ⁺¹+ Z

{ψ<ϕ}

ω∧ω_ϕⁿ⁻¹. From (1), (5) and (6) we obtain

1 (1−)ⁿ¹

h

1−2aⁿ⁺¹−2C1(α, A)aⁿ⁺¹i 6 1

(1−)ⁿ¹ hZ

{|ϕ−ψ|>aⁿ⁺¹}

ωⁿ_ϕ−2C₁(α, A)a^1+αi 64aⁿ⁺¹+ 1.

Hence

61−h1−2(C₁(α, A) + 1)aⁿ⁺¹ 4aⁿ⁺¹+ 1

in

6C₂(α, A)aⁿ⁺¹. This implies that there existst∈Rsatisfying

Z

{|ϕ−ψ−t|>a}

ω_ϕⁿ62C₂(α, A)aⁿ⁺¹.

Finally we have Z

{|ϕ−ψ−t|>a}

(ωⁿ_ϕ+ωⁿ_ψ) = 2 Z

{|ϕ−ψ−t|>a}

ω_ϕⁿ+ Z

{|ϕ−ψ−t|>a}

(ωⁿ_ψ−ω_ϕⁿ) 62C2(α, A)aⁿ⁺¹+a^2n+3+β6C(α, A)aⁿ⁺¹.

The second step in proving our stability theorem is the following 3.2. Proposition. — Let ϕ, ψ ∈ E⁻(X, ω) be such that ωⁿ_ϕ, ωⁿ_ψ ∈ H(α, A). Then there exist constantst∈RandC(α, A)>0such that

C_X({|ϕ−ψ−t|> a})6C(α, A)a, herea= [R

Xkωⁿ_ϕ−ωⁿ_ψk]

1 2n+3+n+1

1+α.

Proof. — Since CX({|ϕ−ψ−t| > a}) 6 CX(X) = 1, it suffices to consider the case whenais small. Without loss of generality we can assume thatsup_Xϕ= sup_Xψ= 0. By Remark 2.5 in [18] there existsM(α, A)>0 such thatkϕk_L∞(X)< M(α, A),kψk_L∞(X)< M(α, A). By Proposition 3.1 we can findt >0 such that

Z

{|ϕ−ψ−t|>a}

(ωⁿ_ϕ+ωⁿ_ψ)6C1(α, A)aⁿ⁺¹. We consider the casea <min(1,_C ¹

1(α,A)). SinceR

{|ϕ−ψ−t|>a}(ω_ϕⁿ+ω_ψⁿ)<1 we get{|ϕ−ψ−t|> a} 6=X. This implies that |t|6sup_X|ϕ−ψ|+ 16 M(α, A)+1. Replacingψbyψ+twe can assume thatt= 0andkψkL^∞(X)<

2M(α, A) + 1. Using Lemma 2.3 in [18] for s=^a₂,t= _{2(2M(α,A)+1)}^a we get CX({ϕ−ψ <−a})6CX

n

ϕ−ψ <−a

2 − a

2(2M(α, A) + 1) o

62ⁿ(2M(α, A) + 1)ⁿ aⁿ

Z

{ϕ−ψ<−a}

ω_ϕⁿ 62ⁿ(2M(α, A) + 1)ⁿC₁(α, A)a.

Similarly we get

CX({ψ−ϕ <−a})62ⁿ(2M(α, A) + 1)ⁿC1(α, A)a.

Combination of these inequalities yields

CX({|ϕ−ψ|> a})6C(α, A)a.

Now we prove the promised generalization of Kołodziej stability theorem

(Theorem 1.1 in [27]).

3.3. Theorem. — Let ϕ, ψ ∈ E⁻(X, ω) be such that sup_Xϕ = sup_Xψ = 0 and ω_ϕⁿ, ωⁿ_ψ ∈ H(α, A). Then there exists C(α, A) > 0 such that

sup

X

|ϕ−ψ|6C(α, A)hZ

X

kω_ϕⁿ−ω_ψⁿki

min(1, αn) 2n+3+n+1

1+α. Proof. — Set

a=hZ

X

kωⁿ_ϕ−ω_ψⁿki ¹

2n+3+n+1 1+α.

By Proposition 3.2 there exists C1(α, A)> 0 and t ∈ R such that |t| 6 M(α, A) + 1and

C_X({|ϕ−ψ−t|> a})6C₁(α, A)a.

Moreover, by Proposition 2.6 in [18] there existsC₂(α, A)>0 such that sup

X

|ϕ−ψ−t|62a+C₂(α, A)[C_X({|ϕ−ψ−t|> a})]^αn 62a+C2(α, A)[C1(α, A)a]^αn

6C3(α, A)a^min(1,αn).

Moreover, since sup_Xϕ= sup_Xψ = 0we obtain |t|6C3(α, A)a^min(1,αn). Combination of these inequalities yields

sup

X

|ϕ−ψ|6sup

X

|ϕ−ψ−t|+|t|62C₃(α, A)a^min(1,αn)

=C(α, A)hZ

X

kωⁿ_ϕ−ω_ψⁿki

min(1, αn) 2n+3+n+1

1+α.

3.4.Corollary. — Letµbe a non-negative Radon measure onXsuch thatµ(B(z, r))6Ar^2n−2+α for allB(z, r)⊂X (A, α >0are constants).

Givenp > 1, M >0, > 0 and f, g∈L^p(dµ)with kfk_Lp(dµ),kgk_Lp(dµ)6 M and R

Xf dµ = R

Xgdµ = 1. Assume that ϕ, ψ ∈ E⁻(X, ω) satisfy ωⁿ_ϕ = f dµ, ω_ψⁿ = gdµ and sup_Xϕ = sup_Xψ = 0. Then there exists C(α, A, M, )>0such that

sup

X

|ϕ−ψ|6C(α, A, M, )hZ

X

|f−g|dµi_2n+3+¹ . Proof. — By Hölder inequality we have

Z

K

f dµ6kfk_Lp(dµ)[µ(K)]^1−p1 6M[µ(K)]^1−p1, Z

K

gdµ6kgk_Lp(dµ)[µ(K)]^1−1p 6M[µ(K)]^1−1p,

for any Borel subsetKofX. By Proposition 2.7 we getf dµ, gdµ∈ H(∞).

Using Theorem 3.3 we can findC(α, A, M, )>0 such that sup

X

|ϕ−ψ|6C(α, A, M, )hZ

X

|f−g|dµi_2n+3+¹ .

4. Local estimates in Potential theory

Let Ωbe a bounded domain in Rⁿ (n>2). By SH(Ω) (resp. SH⁻(Ω)) we denote the set of subharmonic (resp. negative subharmonic) functions onΩ. For eachu∈SH(Ω) andδ >0we denote

˜

u_δ(x) = 1 c_nδⁿ

Z

B_δ

u(x+y)dV_n(y), uδ(x) = sup

y∈Bδ

u(x+y),

forx∈Ωδ={x∈Ω :d(x, ∂Ω)> δ}. HereBδ ={x∈Rⁿ:|x|= (x²₁+· · ·+ x²_n)¹² < δ}andcn is the volume of the unit ball B1. We state some results which will be used in our main theorems.

4.1. Theorem. — Let µbe a non-negative Radon measure on Ωsuch that µ(B(z, r)) 6 Ar^n−2+α for all B(z, r) ⊂ D ⊂⊂ Ω (A, α > 0 are constants). Then forK ⊂⊂ D and > 0 there exists C(α, A, K, ) such

that Z

K

[˜u_δ−u]dµ6C(α, A, K, ) Z

D¯

∆u δ^α−1+α, for allu∈SH(Ω), where∆ is the Laplace operator.

Proof. — Since the change of radii of the balls does not affect the state- ment we can assume that Ω =B₄, D =B₃, K =B₁ and uis smooth on B4. By [22] we have

u(x) = Z

B2

G(x, z)∆u(z) +h(x),

where G(x, y) is the fundamental solution of Laplace equation and h is harmonic inB₂. By Fubini theorem we have

Z

B₁

[˜uδ(x)−u(x)]dµ(x)

= Z

B₁

1 cnδⁿ

Z

B_δ

[u(x+y)−u(x)]dV_n(y)dµ(x)

· 1 cnδⁿ

Z

B₁

Z

B_δ

Z

B₂

[G(x+y, z)−G(x, z)]∆u(z)dV_n(y)dµ(x)

= Z

B2

∆u(z) 1 cnδⁿ

Z

Bδ

dVn(y) Z

B1

[G(x+y, z)−G(x, z)]dµ(x) Set

F(y, z) = Z

B₁

[G(x+y, z)−G(x, z)]dµ(x).

It is enough to prove thatF(y, z)6C(α, A, s)δ^α−1+α for ally∈Bδ, z∈B2. We consider two cases:

Case 1: n= 2. Fory∈Bδ, z∈B2,δ < ¹₂, we have F(y, z) =

Z

B₁

[ln|x+y−z| −ln|x−z|]dµ(x)

= Z

B₁∩{|x−z|>|y|^1+α1 }

ln|1 + y

x−z|dµ(x) + Z

B₁∩{|x−z|<|y|^1+α1 }

ln|1

+ y

x−z|dµ(x) 6

Z

B1∩{|x−z|>|y|^1+α1 }

ln(1 +|y|^1+αα )dµ(x) + ln 4

Z

B₁∩{|x−z|<|y|^1+α1 }

dµ+ Z

B₁∩{|x−z|<|y|^1+α1 }

ln 1

|x−z|dµ(x) 6|y|^1+αα µ(B1) +A|y|^1+αα ln 4

+|y|^α−1+α Z

{|x−z|<|y|^1+α1 }

1

|x−z|^α−ln 1

|x−z|dµ(x) 6A(1 + ln 4)|y|^1+αα +|y|^α−1+αC₁(α, )

Z

{|x−z|<1}

dµ(x)

|x−z|^α−2 6A(1 + ln 4)|y|^1+αα

+C1(α, )|y|^α−1+α

∞

X

j=0

Z

{2^−j−16|x−z|<2^−j}

dµ(x)

|x−z|^α−2

6A(1 + ln 4)|y|^1+αα +C₁(α, )|y|^α−1+αA

∞

X

j=0

2^{(j+1)(α−2)−jα} 6C(α, A, )|y|^α−1+α 6C(α, A, )δ^α−1+α.

Case 2: n>3. Similarly fory∈Bδ, z∈B2, δ < ¹₂, we have F(y, z) =

Z

B₁

h− 1

|x+y−z|ⁿ⁻² + 1

|x−z|ⁿ⁻² idµ(x)

= Z

B₁∩{|x−z|>|y|^1+α1 }

|x+y−z|ⁿ⁻²− |x−z|ⁿ⁻²

|x+y−z|ⁿ⁻²|x−z|ⁿ⁻² dµ(x) +

Z

{|x−z|<|y|^1+α1 }

dµ(x)

|x−z|ⁿ⁻² 6C₂(α)|y|^1+αα

Z

B1∩{|x−z|>|y|^1+α1 }

dµ(x) +|y|^α−1+α

Z

{|x−z|<|y|^1+α1 }

dµ(x)

|x−z|^n−2+α−

6AC₂(α)|y|^1+αα +|y|^α−1+α Z

{|x−z|<1}

dµ(x)

|x−z|^n−2+α−

6C(α, A, )|y|^α−1+α 6C(α, A, )δ^α−1+α,

4.2. Theorem. — Let µbe a non-negative Radon measure on Ωsuch that µ(B(z, r)) 6 Ar^n−2+α for all B(z, r) ⊂ D ⊂⊂ Ω (A, α > 0 are constants). Then forK ⊂⊂ D and > 0 there exists C(α, A, K, ) such that

Z

K

[uδ−u]dµ6C(α, A, K, )kuk_L^∞_(Ω)δ^2(1+α)α− , for allu∈SH∩L^∞(Ω).

We need a well-known lemma:

4.3. Lemma. — Letu∈SH∩L^∞(Ω). Then

|u˜δ(x)−u˜δ(y)|6 kukL^∞(Ω)|x−y|

δ ,

for allx, y∈Ωδ.

Proof. — Proof of Theorem 4.2 By Lemma 4.3 we have uδ(x) = sup

y∈Bδ

u(x+y)6 sup

y∈Bδ

˜

uδ¹2(x+y)6u˜

δ¹2(x) +δ¹²kuk_L^∞_(Ω). By Theorem 4.1 we get

Z

K

[uδ−u]dµ6 Z

K

[˜u

δ¹2 −u]dµ+kuk_L^∞_(Ω)µ(K)δ¹² 6C(α, A, K, )kuk_L^∞_(Ω) δ

α−

2(1+α).

Next we state a well-known result is a direct consequence of the Jensen

formula (see [1])

4.4. Proposition. — Let u ∈ SH(B₂) be such that |u(x)−u(y)| 6 A|x−y|^αfor allx, y∈B2. Then there existsC(α, A)>0such that

Z

B(x,r)

∆u6C(α, A)r^n−2+α, for allB(x, r)⊂B1.

5. Main results

Proof of Theorem A. — We use the same scheme as the proof of Theo- rem 2.1 in [27]. From Corollary 3.4 and from Theorem 4.2 we can replaceωⁿ bydµ. This implies thatuis Hölder continuous with the Hölder exponent dependent onα,A, p,X andkfk_Lp(dµ).

Proof of Corollary B. — It follows from Proposition 4.4 and Theorem A.

Proof of Corollary C. — Direct application of Theorem A.

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Manuscrit reçu le 5 mai 2009, accepté le 21 septembre 2009.

Pham Hoang HIEP

University of Education (Dai hoc Su Pham Ha Noi) Department of Mathematics

CauGiay, Hanoi (Vietnam) [email protected]