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REGULARITY IN MORREY SPACES OF STRONG SOLUTIONS TO NONDIVERGENCE ELLIPTIC

EQUATIONS WITH V M O COEFFICIENTS

DASHAN FAN, SHANZHEN LU, AND DACHUN YANG

Abstract. In this paper, by means of the theories of singular inte-grals and linear commutators,the authors establish the regularity in Morrey spaces of strong solutions to nondivergence elliptic equations withV M Ocoefficients.

1. Introduction

Let Ω be an open set ofRn_,_p_∈₍₁_,_∞_{), and}_λ_∈₍₀_{, n}_{). For}_f _∈_L1

loc(Ω), let

kfkp_Lp,λ_(Ω)= sup ρ>0,x∈Ω

1

ρλ

Z

Bρ(x)∩Ω

|f(y)|p_dy

and define Lp,λ_{(Ω) to be the set of measurable functions} _f _{such that}

kfkLp,λ_(Ω)<∞, where, and in what follows,Bρ(x) ={y∈Rn :|x−y|< ρ}

for anyρ >0.The spaceLp,λ_{(Ω) is usually called the Morrey space.}

Assumingf ∈Lp,λ_{(Ω), the main purpose of this paper is to investigate}

the regularity in the Morrey space of the strong solution to the following Dirichlet problem on the second-order elliptic equation in nondivergence form:

    

Lu≡

n

X

i,j=1

aij(x)uxixj =f a.e. in Ω,

u= 0 on ∂Ω,

(1.1)

where x = (x1, . . . , xn) ∈ Rn; Ω is a bounded domain C1,1 of Rn; the coefficients {aij}n

i,j=1 of L are symmetric and uniformly elliptic, i.e., for 1991Mathematics Subject Classification. 35J25, 42B20.

Key words and phrases. Elliptic equation, Morrey space, V M O, singular integral, commutator.

425

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someν ≥1 and everyξ∈Rn_,

aij(x) =aji(x) and ν−1|ξ|2≤

n

X

i,j=1

aij(x)ξiξj ≤ν|ξ|2 (1.2)

with a.e. x∈ Ω. Moreover, we assume thataij ∈ V M O(Ω), the space of the functions of vanishing mean oscillation introduced by Sarason in [1].

Our method is based on integral representation formulas established in [2, 3] for the second derivatives of the solutionuto (1.1), and on the theories of singular integrals and linear commutators in Morrey spaces. In fact, in

§2, we will establish the boundedness in Morrey spaces for a large class of singular integrals and linear commutators. From this, we can deduce the interior estimates and boundary estimates for the solution to (1.1); therefore, by a standard procedure, we can obtain its whole estimates in Morrey spaces (see [3] and [4]). This will be done in§3.

It is worth pointing out that part of the interior estimates for the solution to (1.1) have been obtained in [5]. Here, we obtain the whole interior esti-mates in a different way, which seems much simpler than the corresponding ones in [5].

2. Singular Integrals and Linear Commutators

First, we have the following general theorem for the boundedness in Mor-rey spaces of sublinear operators.

Theorem 2.1. Let p∈(1,∞)andλ∈(0, n). If a sublinear operatorT

is bounded on Lp₍_Rn₎ _{and for any} _f _∈ _L1₍_Rn₎ _{with compact support and} x6∈suppf,

|T f(x)| ≤c

Z

Rn

|f(y)|

|x−y|ndy, (2.1)

thenT is also bounded on Lp,λ₍_Rn_).

Proof. Fixx∈Rn _and_{r >}_{0. Write}

f(y) =f(y)χB2r(x)(y) + ∞

X

k=1

f(y)χB₂k+1_r(x)\B2k r(x)(y)≡ ∞

X

k=0

fk(y). (2.2)

Thus, by theLp(Rn_{)-boundedness of}_T _{and (2.1), we have}

Z

Br(x)

|T f(z)|pdz

1/p

≤

∞

X

k=0 Z

Br(x)

|T fk(z)|pdz

1/p

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≤cpkf0kLp₍_Rn₎+c ∞

X

k=1 Z

Br(x)

Z

B₂k+1_r(x)\B2k r(x)

|f(y)| |z−y|ndy

p dz

1/p

≤

≤cprλ/pkfkLp,λ₍_Rn₎+cp ∞

X

k=1

rn/p

(2k_r₎n/p

Z

B2k+1_r(x)

|f(y)|pdy

1/p

≤

≤cprλ/pkfkLp,λ₍_Rn₎+cprλ/pkfk_Lp,λ₍_Rn₎ ∞

X

k=1 1 2k(n−λ)/p ≤

≤cprλ/pkfkLp,λ₍_Rn₎.

Therefore,kT fkLp,λ₍_Rn₎≤ckfk_Lp,λ₍_Rn₎.

Condition (2.1) can be satisfied by many operators such as Bochner–Riesz operators at the critical index, Ricci–Stein’s oscillatory singular integral, C. Fefferman’s singular multiplier, and the following Calder´on–Zygmund operators.

Definition 2.1. Letk: Rn_{\ {}₀_{} →}R_{. We say that}_k₍_x_{) is a constant}

Calder´on–Zygmund kernel (constantC−Z kernel) if (i)k∈C∞₍_Rn_{\ {}₀_}_);

(ii)kis homogeneous of degree−n;

(iii)R_Σk(x)dσ= 0,where, and in what follows, Σ ={x∈Rn _:_|_x_|_{= 1}_}_. Definition 2.2. Let Ω be an open subset ofRn_and_k_{: Ω}_×{Rn_\{₀_{}} →} R_{. We say that}_k₍_x_{) is a variable} _C₋_Z _{kernel on Ω if}

(i)k(x,·) is aC−Z kernel fora.e. x∈Ω;

(ii) max|j|≤2nk(∂j/∂zj)k(x, z)kL∞_(Ω_×_Σ)≡M <∞.

Let k be a constant or a variable C−Z kernel on Ω. We define the correspondingC−Z operator by

T f(x) =p.v.

Z

Rn

k(x−y)f(y)dy or T f(x) =p.v.

Z

Ω

k(x, x−y)f(y)dy.

Obviously, in these cases it satisfies the conditions of Theorem 2.1; see Theorems 2.10 and 2.5 in [2]. Thus we have the following simple corollary.

Corollary 2.1. Let p ∈ (1,∞) and λ ∈ (0, n). If k is a constant or a variable C−Z kernel on Rn and T is the corresponding operator, then there exists a constantc =c(n, p, λ, k) orc =c(n, p, λ, k, M)such that for allf ∈Lp,λ₍_Rn_),_k_{T f}_k

Lp,λ₍_Rn₎≤c(p, λ, k)kfk_Lp,λ₍_Rn₎.

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Corollary 2.2. Let p ∈ (1,∞), λ ∈ (0, n), and Ω be an open subset of Rn. If k is a variable C−Z kernel on Ω and T is the corresponding operator, then there exists a constant c =c(n, p, λ, k, M) such that for all

f ∈Lp,λ_(Ω),

kT fkLp,λ_(Ω)≤c(p, λ, k)kfk_Lp,λ_(Ω).

Now, let us consider the boundedness on Morrey spaces of the linear commutator [a, T] defined by [a, T]f = T(af)(x)−a(x)T f(x). First, we recall the definitions of the spaces BM OandV M O. For the properties of these spaces, we refer to [1], [6], and [7].

Definition 2.3. Let Ω be an open subset of Rn_{. We say that any}_f _∈ L1

loc(Ω) is in the spaceBM O(Ω) if sup

ρ>0,x∈Ω 1

|Bρ(x)∩Ω| Z

Bρ(x)∩Ω

|f(y)−fB_ρ(x)∩Ω|dy≡ kfk∗<∞,

wherefBρ(x)∩Ωis the average over Bρ(x)∩Ω off. Moreover, for anyf ∈BM O(Ω) and r >0,we set

sup

ρ≤r,x∈Ω 1

|Bρ(x)∩Ω| Z

Bρ(x)∩Ω

|f(y)−fBρ(x)∩Ω|dy≡η(r). (2.3)

We say that any f ∈ BM O(Ω) is in the space V M O(Ω) if η(r) → 0 as

r→0 and refer to η(r) as theV M Omodulus of f.

Theorem 2.2. Letp∈(1,∞), λ∈(0, n)anda∈BM O(Rn_{). If a linear}

operatorT satisfies(2.1)and[a, T]is bounded onLp₍_Rn_{), then}_[_{a, T}_]_{is also}

bounded onLp,λ₍_Rn_).

Proof. For any x ∈ Rn _and _{r >} ₀_, _{we write} _f _{as in (2.2). By the} _Lp

-boundedness of [a, T] and (2.1), we obtain Z

Br(x)

[a, T]f(z)pdz

1/p

≤

∞

X

k=0 Z

Br(x)

[a, T]fk(z)pdz

1/p

≤

≤ckf0kLp₍_Rn₎+

+c ∞

X

k=1 Z

Br(x)

Z

B2k+1_r(x)\B2k r(x)

|a(y)−a(z)| |f(y)| |z−y|n dy

p dz

1/p

≤

≤crλ/pkfkLp,λ₍_Rn₎+

+c ∞

X

k=1 1 (2k_r₎n

Z

Br(x)

Z

B₂k+1_r(x)

|a(y)−a(z)| |f(y)|dy

p dz

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For anyσ >0,we set

aσ= 1

|Bσ(x)| Z

Bσ(x)

a(y)dy.

By the John–Nirenberg’s lemma on BM O functions and the fact that

|a2k+1_r−ar| ≤c(n)(k+ 1)kak_∗ (see [7]), we obtain

Z

Br(x)

Z

B₂k+1_r(x)

|a(y)−a(z)| |f(y)|dy

p dz

1/p

≤

≤ Z

Br(x)

|ar−a(z)|pdz

1/p Z

B2k+1_r(x)

|f(y)|dy+

+crn/p

Z

B₂k+1_r(x)

|a(y)−ar| |f(y)|dy≤

≤crn+λ/p2k{n(1−1/p)+λ/p}kak∗kfkLp,λ₍_Rn₎+

+crn/p

Z

B2k+1_r(x)

|a(y)−ar|p′dy

1/p′ Z

B2k+1_r(x)

|f(y)|pdy

1/p

≤

≤crn+λ/p2k{n(1−1/p)+λ/p}kak∗kfkLp,λ₍_Rn₎crn+λ/p2k{n(1−1/p)+λ/p}×

×

₁

|B2k+1_r(x)|

Z

B2k+1_r(x)

|a(y)−a2k+1_r|p ′

dy

1/p′ +

+|a2k+1_r−ar|

kfkLp,λ₍_Rn₎≤

≤c(k+ 1)rn+λ/p2k{n(1−1/p)+λ/p}kak∗kfkLp,λ₍_Rn₎,

where, and in what follows, 1/p+ 1/p′_{= 1}_._Thus,

Z

Br(x)

|[a, T]f(z)|p_dz

1/p

≤

≤crλ/p_k_a_k

∗kfkLp,λ₍_Rn₎

1 +

∞

X

k=1

k+ 1 2k(n−λ)/p

≤crλ/p_k_a_k

∗kfkLp,λ₍_Rn₎.

Therefore

k[a, T]fkLp,λ₍_Rn₎≤ckak_∗kfk_Lp,λ₍_Rn₎.

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Corollary 2.3. Let p ∈ (1,∞), λ ∈ (0, n), and a ∈ BM O(Rn_{). If} _k

is a constant or a variable C−Z kernel on Rn and T the corresponding operator, then there exists a constantc=c(n, p, λ, k)orc=c(n, p, λ, k, M) such that for all f ∈Lp,λ₍_Rn_),

k[a, T]fkLp,λ₍_Rn₎≤ckak∗kfkLp,λ₍_Rn₎.

From this and the extension theorem of BM O(Ω)-functions in [8, page 42 and 54], by a procedure similar to Theorem 2.11 in [2] and Theorem 2.2 in [5], we can obtain the following corollary.

Corollary 2.4. Let p ∈ (1,∞) and λ ∈ (0, n). Suppose Ω is an open subset ofRn_and_a_∈_{BM O}₍Rn_{). If}_k_{is a variable}_C₋_Z _{kernel on}_Ω_and_T

the corresponding operator, then there exists a constant c=c(n, p, λ, k, M) such that for all f ∈Lp,λ_(Ω),

k[a, T]fkLp,λ_(Ω)≤ckak∗kfkLp,λ_(Ω).

We can also have the following local version of Corollary 2.4; see Theorem 2.13 in [1] for the proof.

Corollary 2.5. Let p∈(1,∞)and λ∈(0, n). Let Ωbe an open subset of Rn_{, a}_∈_{V M O}_{(Ω), and}_η _{be as in} ₍₂_._{3). If}_k _{is a variable} _C₋_Z _kernel

on Ω and T the corresponding operator, then for any ε > 0, there exists positive ρ0 =ρ0(ε, η)such that for any ball Br with the radiusr∈(0, ρ0),

Br∩Ω≡Ωr6=∅and allf ∈Lp,λ(Ωr),

k[a, T]fkLp,λ_(Ω_r₎≤cεkfk_Lp,λ_(Ω_r₎,

wherec=c(a, p, λ, k)is independent of f andε.

It is worth pointing out that Corollaries 2.4 and 2.5 have been obtained by Fazio and Ragusa in [5] in a different way. It seems that our method is much simpler than theirs.

Let Rn₊ ₌ _{_x _{= (}_x′_{, xn}_{) :} _x′ _{= (}_x₁_{, . . . , xn−}₁₎ _∈ Rn−1_{, xn} _> ₀_}_{. To}

establish the boundary estimates of the solutions to (1.1), we need to study the boundedness on Lp,λ₍_Rn

+) of some other integral operators. First, we have the following general theorem for sublinear operators.

Theorem 2.3. Let p ∈ (1,∞), λ ∈ (0, n), and ex= (x′_,₋_xn₎ _for _x₌

(x′_{, xn}₎_∈_Rn

+. If a sublinear operatorT is bounded onLp(Rn+)and for any

f ∈L1₍_Rn

+)with compact support andx∈Rn+,

|T f(x)| ≤c

Z

Rn

+

|f(y)|

|_ex−y|ndy, (2.4)

thenT is also bounded onLp,λ₍_Rn

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Proof. Letz∈Rn₊_and_{σ >}₀_._{In what follows, we set}_B_σ+₍_z_{) =}_Bσ₍_z₎_∩Rn₊_.

We consider two cases.

Case 1. 0≤zn<2σ. In this case, we write

f(y) =f(y)χ_B+ 24σ(z)

(y) +

∞

X

ℓ=4

f(y)χ_B+

2ℓ+1_σ(z)\B +

2ℓ σ(z)(y)≡ ∞

X

ℓ=3

fℓ(y). (2.5)

Therefore, byLp_{-boundedness of}_T _{and (2.4), we obtain}

Z

Bσ+(z)

|T f(x)|pdx

1/p

≤

∞

X

ℓ=3 Z

B+σ(z)

|T fℓ(x)|pdx

1/p

≤

≤ckf3kLp₍_Rn

+)+c

∞

X

ℓ=4 Z

B+

σ(z)

Z

B+

2ℓ+1σ(z)\B

+ 2ℓ σ(z)

|f(y)| |x_e−y|n dy

p dx

1/p

≤

≤cσλ/pkfkLp,λ₍_Rn

+)+c

∞

X

ℓ=4 1 (2ℓ_σ₎n

Z

B+

σ(z)

Z

B+ 2ℓ+1_σ(z)

|f(y)|dy

p dx

1/p

≤

≤cσλ/p_k_f_k Lp,λ₍_Rn

+) 1 + ∞ X ℓ=4 1 2ℓ(n−λ)/p

≤cσλ/p_k_f_k Lp,λ₍_Rn

+),

which is the desirable estimate.

Case 2. There existsℓ∈N_{such that 2}ℓ_σ_≤_zn_<₂ℓ+1_σ_{. In this case, we}

write

f(y) =f(y)χ_B+

2ℓ+4_σ(z)(y) +

∞

X

k=1

f(y)χ_B+

2ℓ+k+4_σ(z)\B +

2ℓ+k+3_σ(z)(y)≡

≡

∞

X

k=0

fk(y). (2.6)

From (2.4), it follows that

Z

B+

σ(z)

|T f(x)|p_dx

1/p

≤c

Z

B+

σ(z)

Z

B+ 2ℓ+4_σ(z)

|f(y)| |xe−y|n dy

p dx

1/p

+ +c ∞ X k=1 Z B+

σ(z)

Z

B+

2ℓ+k+4σ(z)\B

+

2ℓ+k+3σ(z)

|f(y)| |x_e−y|ndy

p dx

1/p

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≤ c

(2ℓ_σ₎n

Z

B+

σ(z)

Z

B+ 2ℓ+4_σ(z)

|f(y)|dypdx

1/p

+

+c ∞

X

k=1 1 (2ℓ+k_σ₎n

Z

B+

σ(z)

Z

B+

2ℓ+k+4_σ(z)

|f(y)|dy

p dx

1/p

≤

+)

1 2ℓ(n−λ)/p +

∞

X

k=1 1 2(ℓ+k)(n−λ)/p

+), by noting thatℓ∈N _and_{λ < n}_{. From this, it is easy to deduce Theorem}

2.3.

To state the following corollary, we need more notation. Let a(x) =

{ain(x)}n

i=1be as in (1.2) and define

T(x, y)≡x− 2xn

ann(y)a(y).

Then, as a simple corollary of the above Theorem 2.3 and Lemma 3.1 in [3], we have

Corollary 2.6. Let p∈(1,∞), λ∈(0, n), and Ω be an open subset of

Rn₊_{. If}_k_{is a variable}_C₋_Z _{kernel on}_{Ω, and for} _x_∈_Ω_{we define}

e

T f(x) = Z

Ω

k(x, T(x)−y)f(y)dy

with T(x) =T(x, x), then there exists a constantc=c(p, λ, ν, k)such that for allf ∈Lp,λ_(Ω),

kT fe kLp,λ_(Ω)≤ckfk_Lp,λ_(Ω).

For the linear commutator onRn

+,we have

Theorem 2.4. Let p ∈ (1,∞), λ ∈ (0, n), and a ∈ BM O(Rn₊_{). If a}

linear operatorTesatisfies(2.4)and[a,Te]is bounded onLp₍_Rn

+), then[a,Te] is also bounded onLp,λ₍_Rn

+).

Proof. Letz ∈ Rn₊ _and _{σ >} _{0. Similarly to the proof of Theorem 2.3, we}

also consider two cases.

Case1. 0≤zn <2σ. In this case, we writef as in (2.5). We then have Z

B+

σ(z)

|[a,Te]f(x)|pdx

1/p

≤

∞

X

j=3 Z

B+

σ(z)

|[a,Te]fj(x)|pdx

1/p

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≤ckak∗kf3kLp₍_Rn +)+ +c ∞ X j=4 Z B+

σ(z)

Z

B+

2j+1σ(z)\B

+ 2j σ(z)

|a(y)−a(x)||f(y)| |xe−y|n dy

p dx

1/p

≤

≤cσλ/pkak∗kfkLp,λ₍_Rn

+)+ +c ∞ X j=4 1 (2j_σ₎n

Z

B+

σ(z)

Z

B+ 2j+1_σ(z)

|a(y)−a(x)| |f(y)|dy

p dx

1/p

≤

≤cσλ/p_k_a_k

∗kfkLp,λ₍_Rn

+)+c

∞

X

j=4 1 (2j_σ₎n

Z

B+

σ(z)

|aσ−a(x)|p_dx

1/p

×

Z

B+ 2j+1σ(z)

|f(y)|dy

+ +c ∞ X j=4 1 (2j_σ₎n

Z

B+

σ(z)

Z

B+ 2j+1_σ(z)

|a(y)−aσ| |f(y)|dy

p dx

1/p

≤

+)+cσ

λ/p_k_a_k

∗kfkLp,λ₍_Rn

+)

_X∞

j=4 1 2j(n−λ)/p

+ +c ∞ X j=4 1 (2j_σ₎n(2

j_σ₎λ/p_k_f_k Lp,λ₍_Rn

+)σ

n/p

Z

B+ 2j+1_σ(z)

|a(y)−aσ|p′dy

1/p′

≤

≤cσλ/p_k_a_k

∗kfkLp,λ₍_Rn

+)+cσ

n/p_k_f_k Lp,λ₍_Rn

+)

_X∞

j=4 1

(2j_σ₎(n−λ)/p(2 j_σ₎n/p′

×

₁

|B₂+j+1_σ(z)|

Z

B+ 2j+1σ(z)

|a(y)−a2j+1_σ|p ′

dy

1/p′

+|a2j+1_σ−aσ|

≤

+)+

+cσλ/pkfkLp,λ₍_Rn

+)

∞

X

j=4 1 2j(n−λ)/p

kak∗+|a2j+1_σ−aσ|

≤

≤cσλ/p_k_a_k

∗kfkLp,λ₍_Rn

+)

_X∞

j=4

j+ 1 2j(n−λ)/p

≤cσλ/p_k_a_k

∗kfkLp,λ₍_Rn

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where for anyr >0 we set

ar= 1

|B+r(x)|

Z

B+

r(x) a(y)dy

and use the fact thatλ < nand|a2j+1_σ−aσ| ≤c(n)jkak_∗. This estimation is the expected one.

Case 2. There existsℓ∈N_{such that 2}ℓ_σ_≤_zn_<₂ℓ+1_σ_{. In this case, we}

writef as in (2.6). We then have Z

B+

σ(z)

|[a,Te]f(x)|p_dx

1/p

≤

∞

X

k=0 Z

B+

σ(z)

|[a,Te]fk(x)|p_dx

1/p

≤

≤c

Z

B+

σ(z)

Z

B+ 2ℓ+4_σ(z)

|a(y)−a(x)||f(y)| |ex−y|n dy

p dx

1/p

+

+c ∞

X

k=1 Z

B+

σ(z)

Z

B+

2k+ℓ+4σ(z)\B

+

2k+ℓ+3σ(z)

|a(y)−a(x)||f(y)| |x_e−y|n dy

p dx

1/p

≤

≤ c

(2ℓ_σ₎n

n Z

B+

σ(z)

Z

B+ 2ℓ+4_σ(z)

|a(y)−a(x)||f(y)|dy

p dx

1/p

+

+c ∞

X

k=1 1 (2k+ℓ_σ₎n

Z

B+

σ(z)

Z

B+

2k+ℓ+4_σ(z)

|a(y)−a(x)| |f(y)|dy

p dx

1/p

≤

+)

_X∞

k=1 1 2(k+ℓ)(n−λ)/p

+),

sinceℓ ∈N _and_{λ < n}_{. Here, in the next to the last inequality, we used a}

computation technique similar to Case 1. By the above estimate we easily finish the proof of Theorem 2.4.

We can also obtain the local version of Theorem 2.4; see Theorem 2.13 in [2] for the proof.

Corollary 2.7. Let p∈(1,∞),λ∈(0, n), and for anyσ >0,B+

σ(x) =

{(x′_{, xn}₎ _∈_Rn _: _|_x_| _{< σ, xn} _>₀_}_{. Let} _a_∈_{V M O}₍_Rn

+) andη be its V M O modulus. If k is a variable C−Z kernel on Rn₊ _and _Te _{is as in Corollary}

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any r∈(0, ρ0) andf ∈Lp,λ₍_B+

r),

k[a,Te]fk_Lp,λ₍_B+

r)≤cεkfkLp,λ(B

+

r)

withc=c(ν, p, λ, k)independent of f,ε, andr.

3. Elliptic Equations with V M O Coefficients

In this section, we will establish the regularity of the solution to (1.1). First, we have the following definition.

Definition 3.1. Letp∈(1,∞), λ∈(0, n), and Ω be an open subset of

Rn_. _f _∈_L1

loc(Ω) is said to belong to the Sobolev–Morrey spaceW2Lp,λ(Ω) if and only ifuand its distributional derivatives,uxi, uxixj (i, j= 1, . . . , n) are inLp,λ_{(Ω). Moreover, let}_k_u_k

W2_Lp,λ_(Ω)≡ kuk_Lp,λ_(Ω)+Pn_i₌₁kuxik_Lp,λ_(Ω)

+Pn_i,j₌₁kuxixjkLp,λ_(Ω).

We also assume thatf ∈W2

locLp,λ(Ω) iff ∈W2Lp,λ(Ω′) for every Ω′ ⊂⊂

Ω.

Now, let Ω be an open bounded subset ofRn _with_n_≥_{3 and}_∂_Ω_∈_C1,1_,

L≡

n

X

i,j=1

aij(x) ∂ 2

∂xi∂xj

with aij’s satisfying (1.2). We also assume that aij ∈ V M O(Ω). Since for each function in V M O(Ω) there is an extension to Rn _{with the} _{V M O}

modulus controlled by its original one (see [8, page 42 and 54]), without loss of generality, we may assume that aij’s belong to V M O(Rn_{). Let} f ∈Lp,λ_(Ω),_p_∈₍₁_,_∞_{) and}_λ_∈₍₀_{, n}_{). We are interested in the following}

Cauchy problem: (

Lu=f a.e. in Ω, u∈W2_Lp,λ_(Ω)_∩_W1,p

0 (Ω).

(3.1)

Note that Ω is bounded; therefore f ∈Lp,λ_{(Ω) implies} _f _∈_Lp_{(Ω). By the}

results in [3], we know that forf ∈Lp,λ(Ω) with p∈(1,∞) andλ∈(0, n),

(3.1) has a unique solutionu∈W2,p_(Ω)_∩_W1,p

0 (Ω) satisfying

kukW2,p_(Ω)≤ckfk_Lp_(Ω), (3.2)

wherecis independent off. Our main interest here is to improve (3.2) into

kukW2_Lp,λ_(Ω)≤ckfk_Lp,λ_(Ω). (3.3)

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Theorem 3.1. LetLsatisfy the above assumption andη= (Pn_i,j₌₁η2

ij)1/2

where ηij is the V M Omodulus of aij in Ω. Let λ∈(0, n), q, p∈(1,∞), q≤p,f ∈Lp,λ_(Ω)_{, u}_∈_W2_Lq,λ_(Ω)_∩_W1,q

0 (Ω), andLu=f a.e.inΩ.Then

u ∈ W2

locLp,λ(Ω). Moreover, given any Ω′ ⊂⊂ Ω, there exists a constant c=c(n, p, λ, ν,dist(Ω′_{, ∂}_Ω)_{, η}₎_{such that}

kukW2_Lp,λ_(Ω′₎≤c

kukLp,λ_(Ω)+kfk_Lp,λ_(Ω).

The caseq=pof Theorem 3.1 is just Theorem 3.3 in [5]. See [5] for the proof of Theorem 3.1, and we omit the details here. To finish the proof of (3.3), we still need to establish the following boundary estimate.

Theorem 3.2. Let L, λ, q, p, and η be as in Theorem 3.1. Let f ∈

Lp,λ_(Ω)_{, u} _∈ _W2_Lq,λ_(Ω)_∩_W1,q

0 (Ω), and Lu = f a.e. in Ω. Then u ∈

W2_Lp,λ_(Ω)_{and there exists a constant} _c₌_c₍_{n, p, λ, ν, ∂}_Ω_{, η}₎_{such that}

kukW2_Lp,λ_(Ω)≤ckuk_Lp,λ_(Ω)+kfk_Lp,λ_(Ω).

To prove Theorem 3.2, we need to introduce more notation. We let

W2,p γ0 (B

+

σ) be the closure in W2,pof the subspace Cγ0=

n

u: u is the restriction to B+σ

of a function in C0∞(Bσ+) and u(x′,0) = 0

o

,

where, and in what follows,Bσ={x∈Rn _:_|_x_|_{< σ}_}_and _B_σ+₌_{₍_x′_{, xn}₎_∈ Rn _: _|_x_| _{< σ} _and _xn _> ₀_} _{for any} _{σ >} _{0. We also make the following}

assumption and refer to it as assumption (H). 

       

Let n≥3, bij ∈V M O(Rn₎_{, i, j}_{= 1}_{, . . . , n,} bij(x) =bji(x), i, j= 1, . . . , n, a.e. in B+

σ.

There exists µ >0 such that for all ξ∈Rn_, µ−1|ξ|2≤Pni,j=1bij(x)ξiξj≤µ|ξ|2, a.e. in Bσ+.

(H)

Let

e

L≡

n

X

i,j=1

bij ∂

2

∂xi∂xj

and

Γ(x, t) = 1

(n−2)ωn(detbij)1/2 _Xn

i,j=1

Bij(x)titj(2−n)/2,

Γi(x, t) = ∂

∂tiΓ(x, t), Γij(x, t) = ∂2

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for a.e. x ∈ B+

σ and all t ∈ Rn \ {0}, where Bij’s are the entries of the

inverse of the matrix{bij}i,j=1,...,n. We also setb(x) ={bin(x)}ni=1and

T(x, y) =x− 2xn

bnn(y)b(y).

By a covering and flattening argument, the proof of Theorem 3.2 can be reduced to establishing the estimate in Lp,λ₍_B+

r) of uxixj with u ∈ W2,q

γ0 (B +

r) and Lue ∈ Lp,λ(Br+), where 1 < q ≤ p < ∞ and λ ∈ (0, n).

To do so, we need the following key lemma established in [3, page 847].

Lemma 3.1. Assume(H)and letu∈Wγ20,p(B +

σ)withp∈(1,∞). Then uxixj(x) =

=p.v.

Z

B+

σ

Γij(x, x−y)

_Xn

h,k=1

[bhk(x)−bhk(y)]uxhxk(y) +Lue (y)

dy

+Lue (x) Z

|t|=1

Γi(x, y)tjdσt+Iij(x), (3.4)

where fori, j= 1, . . . , n−1,

Iij(x) =

=p.v.

Z

B+

σ

Γij(x, T(x)−y)

_Xn

h,k=1

[bhk(x)−bhk(y)]uxhxk(y) +Lue (y)

dy;

fori= 1, . . . , n−1,

Iin(x) =Ini(x) = Z

B+

σ

_Xn

ℓ=1

Bℓ(x)Γiℓ(x, T(x)−y)

{· · · }dy,

and

Inn(x) = Z

B+

σ n

X

ℓ,k=1

Bℓ(x)Bk(x)Γℓk(x, T(x)−y){· · · }dy;

in the formulas above T(x) = T(x;x), Bi(x) is the i-th component of the vector B(x) =T(en;x)with en = (0, . . . ,0,1), tj is the j-th component of the outer normal to the sphere Σ, and in the curly brackets there is always the same expression as in the first case.

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Theorem 3.3. Assume (H). Let λ ∈ (0, n), q, p ∈ (1,∞), and q ≤ p. Set _eη= (

n

P

i,j=1e

η2

ij)1/2, whereηije is theV M Omodulus of bij,and

M ≡ max

i,j=1,...,n|α|≤max2n

∂

α

∂tαΓij(x, t)

_L∞₍_B+

σ×Σ) .

Then there exists a positive numberρ0=ρ0(n, q, p, M, µ,η, λe ),ρ0< σ,such that for anyr∈(0, ρ0)and anyu∈W2,q

γ0 (B +

r)withLue ∈Lp,λ(B+r), we have u∈W2_Lp,λ₍_B+

r). Furthermore, there exists a constantc=c(n, p, λ, M, µ,ηe)

such that

kuxixjk_Lp,λ₍_B+

r)≤ckLue kLp,λ(B

+

r). (3.5)

Proof. Set fori, j, h, k= 1, . . . , n

Sijhk(f)(x) =p.v.

Z

B+

r

Γij(x, x−y)(bhk(x)−bhk(y))f(y)dy,

and fori, j= 1, . . . , n−1,h, k= 1, . . . , n

e

Sijhk(f)(x) = Z

B+

r

Γij(x, T(x)−y)(bhk(x)−bhk(y))f(y)dy,

fori= 1, . . . , n−1, h, k= 1, . . . , n

e

Sinhk(f)(x) = Z

Br+

_Xn

j=1

Γij(x, T(x)−y)Bj(x)

(bhk(x)−bhk(y))f(y)dy,

and, finally, forh, k= 1, . . . , n

e

Snnhk(f)(x) =

= Z

B+

r

_Xn

i,j=1

Γij(x, T(x)−y)Bi(x)Bj(x)

(bhk(x)−bhk(y))f(y)dy,

wherer∈(0, σ] andf ∈Ls,λ₍_B+

r).

By Lemma 3.1 in [2] and Corollaries 2.5 and 2.7, we can fix ρ0 so small that

X

i,j,h,k

kSijhk+Sijhke k<1,

where the norm of operatorsSijhk+Sijhke is the norm in the space of linear operators from Ls,λ₍_B+

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Consideru∈W2,p γ0 (B

+

r) withLue ∈Lp,λ(Br+),r∈(0, ρ0) and set

hij =p.v.

Z

B+

r

Γij(x, x−y)Lue (y)dy+Lue (x)

Z

|t|=1

Γi(x, t)tjdσt+Iije (x),

where

e

Iij(x) =                     

R

B+

r

Γij(x, T(x)−y)Lue (y)dy for i, j= 1, . . . , n−1,

R

B+

r

_Pn

ℓ=1

Γiℓ(x, T(x)−y)Bℓ(x)

e

Lu(y)dy

for i= 1, . . . , n−1, j=n,

R

Br+

_Pn

ℓ,m=1

Γℓm(x, T(x)−y)Bℓ(x)Bm(x)

e

Lu(y)dy for i=j=n.

From Corollary 2.6, we easily deduce thathij∈Lp,λ₍_B+

r).

Considerw∈[Lp,λ₍_B+

r)]n

2

and defineT w: [Lp,λ₍_B+

r)]n

2

→[Lp,λ₍_B+

r)]n

2

by setting

T w= ((T w)ij)i,j=1,...,n=

_Xn

h,k=1

(Sijhk+Sijhke )(wij) +hij

i,j=1,...,n.

The operatorT is a contraction on [Lp,λ₍_B+

r)]n

2

and thus has a unique fixed pointw_e. Since, by (3.4),{uxixj}i,j=1,...,nis also a fixed point in [Lq,λ(Br+)]n

2

andLp,λ₍_B+

r)⊆Lq,λ(Br+) ifq≤p, the uniqueness of the fixed point implies

thatuxixj =wij_e ∈Lp,λ₍_B+

r) fori, j= 1, . . . , n. Then (3.5) can be deduced

from (3.4), and Corollaries 2.2, 2.4, 2.6, and 2.7.

Acknowledgment

Shanzhen Lu and Dachun Yang are supported by the NNSF of China. Dachun Yang wants to express his thanks to Dr. Guangwei Yuan for many helpful discussions of this subject.

References

1. D. Sarason, On Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207(1975), 391–405.

2. F. Chiarenza, M. Frasca, and P. Longo, Interior W2,p _{estimates for}

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3. F. Chiarenza, M. Frasca, and P. Longo,W2,p_{-solvability of the}

Dirich-let problem for nondivergence elliptic equations with V M O coefficients. Trans. Amer. Math. Soc. 336(1993), 841–853.

4. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. 2nd Edition, Springer–Verlag, Berlin–Heidelberg–New York, 1983.

5. G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112(1993), 241–256.

6. J. B. Garnett, Bounded analytic functions. Academic Press, New York, 1981.

7. A. Torchinsky, Real variable methods in harmonic analysis. Academic Press, New York, 1986.

8. P. W. Jones, Extension theorems forBM O. Indiana Univ. Math. J.

29(1980), 41–66.

(Received 25.12.1996)

Authors’ addresses:

Dashan Fan

Department of Mathematical Sciences University of Wisconsin-Milwaukee P. O. Box 413, Milwaukee, WI 53201 USA

Shanzhen Lu and Dachun Yang Department of Mathematics Beijing Normal University Beijing 100875