GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 5, 1998, 425-440

REGULARITY IN MORREY SPACES OF STRONG SOLUTIONS TO NONDIVERGENCE ELLIPTIC EQUATIONS WITH V MO COEFFICIENTS

DASHAN FAN, SHANZHEN LU, AND DACHUN YANG

Abstract. In this paper, by means of the theories of singular integrals and linear commutators,the authors establish the regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with V MO coefficients.

1. Introduction

Let Ω be an open set of ℝ^{n}, p ∈ (1, ∞), and λ ∈ (0, n). For f ∈ L_{loc}^{1}(Ω), let

∥f∥_{L^{p,λ}(Ω)}^{p} = sup_{ρ>0,x∈Ω} \frac{1}{ρ^{λ}} ∫_{B_{ρ}(x)∩Ω} |f(y)|^{p} dy

and define L^{p,λ}(Ω) to be the set of measurable functions f such that ∥f∥_{L^{p,λ}(Ω)} < ∞, where, and in what follows, B_{ρ}(x) = {y ∈ ℝ^{n} : |x−y| < ρ} for any ρ > 0. The space L^{p,λ}(Ω) is usually called the Morrey space.

Assuming f ∈ L^{p,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:

\begin{cases}Lu ≡ ∑_{i,j=1}^{n} a_{ij}(x)u_{x_{i}x_{j}} = f & a.e. in Ω, \\ u = 0 & on ∂Ω, \end{cases}  (1.1)

where x = (x_{1}, …, x_{n}) ∈ ℝ^{n}; Ω is a bounded domain C^{1,1} of ℝ^{n}; the coefficients {a_{ij}}_{i,j=1}^{n} of L are symmetric and uniformly elliptic, i.e., for

425 some ν ≥ 1 and every ξ ∈ ℝ^{n},

1991 Mathematics Subject Classification. 35J25, 42B20.

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

1072-947X/98/0900-0425$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

a_{ij}(x) = a_{ji}(x) and ν^{−1}|ξ|^{2} ≤ ∑_{i,j=1}^{n} a_{ij}(x)ξ_{i}ξ_{j} ≤ ν|ξ|^{2}  (1.2)

with a.e. x ∈ Ω. Moreover, we assume that a_{ij} ∈ V MO(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 solution u to (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 estimates 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 Morrey spaces of sublinear operators.

Theorem 2.1. Let p ∈ (1, ∞) and λ ∈ (0, n). If a sublinear operator T is bounded on L^{p}(ℝ^{n}) and for any f ∈ L^{1}(ℝ^{n}) with compact support and x ∉ supp f,

|T f(x)| ≤ c ∫_{ℝ^{n}} \frac{|f(y)|}{|x − y|^{n}} dy,  (2.1)

then T is also bounded on L^{p,λ}(ℝ^{n}).

Proof. Fix x ∈ ℝ^{n} and r > 0. Write

f(y) = f(y)χ_{B_{2r}(x)}(y) + ∑_{k=1}^{∞} f(y)χ_{B_{2^{k+1}r}(x)\B_{2^{k}r}(x)}(y) ≡ ∑_{k=0}^{∞} f_{k}(y). (2.2)

Thus, by the L^{p}(ℝ^{n})-boundedness of T and (2.1), we have

( ∫_{B_{r}(x)} |T f(z)|^{p} dz)^{1/p} ≤ ∑_{k=0}^{∞} ( ∫_{B_{r}(x)} |T f_{k}(z)|^{p} dz)^{1/p} ≤

≤ c_{p}∥f_{0}∥_{L^{p}(ℝ^{n})} + c ∑_{k=1}^{∞} \{ ∫_{B_{r}(x)} ( ∫_{B_{2^{k+1}r}(x)\B_{2^{k}r}(x)} \frac{|f(y)|}{|z − y|^{n}} dy)^{p} dz\}^{1/p} ≤

≤ c_{p}r^{λ/p}∥f∥_{L^{p,λ}(ℝ^{n})} + c_{p} ∑_{k=1}^{∞} \frac{r^{n/p}}{(2^{k}r)^{n/p}} ( ∫_{B_{2^{k+1}r}(x)} |f(y)|^{p} dy)^{1/p} ≤

≤ c_{p}r^{λ/p}∥f∥_{L^{p,λ}(ℝ^{n})} + c_{p}r^{λ/p}∥f∥_{L^{p,λ}(ℝ^{n})} ∑_{k=1}^{∞} \frac{1}{2^{k(n−λ)/p}} ≤

≤ c_{p}r^{λ/p}∥f∥_{L^{p,λ}(ℝ^{n})}.

Therefore, ∥T f∥_{L^{p,λ}(ℝ^{n})} ≤ c∥f∥_{L^{p,λ}(ℝ^{n})}.

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 Calderón–Zygmund operators.

Definition 2.1. Let k : ℝ^{n} \ {0} → ℝ. We say that k(x) is a constant Calderón–Zygmund kernel (constant C − Z kernel) if

(i) k ∈ C^{∞}(ℝ^{n} \ {0});

(ii) k is homogeneous of degree −n;

(iii) ∫_{Σ} k(x) dσ = 0, where, and in what follows, Σ = {x ∈ ℝ^{n} : |x| = 1}.

Definition 2.2. Let Ω be an open subset of ℝ^{n} and k : Ω×{ℝ^{n} \{0}} → ℝ. We say that k(x) is a variable C − Z kernel on Ω if

(i) k(x, ·) is a C − Z kernel for a.e. x ∈ Ω;

(ii) max_{|j|≤2n} ∥(∂^{j}/∂z^{j})k(x, z)∥_{L^{∞}(Ω×Σ)} ≡ M < ∞.

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

T f(x) = p.v. ∫_{ℝ^{n}} k(x − y)f(y) dy or T f(x) = p.v. ∫_{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 ℝ^{n} and T is the corresponding operator, then there exists a constant c = c(n, p, λ, k) or c = c(n, p, λ, k, M) such that for all f ∈ L^{p,λ}(ℝ^{n}), ∥T f∥_{L^{p,λ}(ℝ^{n})} ≤ c(p, λ, k)∥f∥_{L^{p,λ}(ℝ^{n})}.

From this corollary, by a proof similar to that of Theorem 2.11 in [2] (see also Theorem 2.2 in [5]), we obtain the following corollary.

Corollary 2.2. Let p ∈ (1, ∞), λ ∈ (0, n), and Ω be an open subset of ℝ^{n}. 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 ∈ L^{p,λ}(Ω),

∥T f∥_{L^{p,λ}(Ω)} ≤ c(p, λ, k)∥f∥_{L^{p,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 BMO and V MO. For the properties of these spaces, we refer to [1], [6], and [7].

Definition 2.3. Let Ω be an open subset of ℝ^{n}. We say that any f ∈ L_{loc}^{1}(Ω) is in the space BMO(Ω) if

sup_{ρ>0,x∈Ω} \frac{1}{|B_{ρ}(x) ∩ Ω|} ∫_{B_{ρ}(x)∩Ω} |f(y) − f_{B_{ρ}(x)∩Ω}| dy ≡ ∥f∥_{∗} < ∞,

where f_{B_{ρ}(x)∩Ω} is the average over B_{ρ}(x) ∩ Ω of f.

Moreover, for any f ∈ BMO(Ω) and r > 0, we set

sup_{ρ≤r,x∈Ω} \frac{1}{|B_{ρ}(x) ∩ Ω|} ∫_{B_{ρ}(x)∩Ω} |f(y) − f_{B_{ρ}(x)∩Ω}| dy ≡ η(r).  (2.3)

We say that any f ∈ BMO(Ω) is in the space V MO(Ω) if η(r) → 0 as r → 0 and refer to η(r) as the V MO modulus of f.

Theorem 2.2. Let p ∈ (1, ∞), λ ∈ (0, n) and a ∈ BMO(ℝ^{n}). If a linear operator T satisfies (2.1) and [a, T] is bounded on L^{p}(ℝ^{n}), then [a, T] is also bounded on L^{p,λ}(ℝ^{n}).

Proof. For any x ∈ ℝ^{n} and r > 0, we write f as in (2.2). By the L^{p}- boundedness of [a, T] and (2.1), we obtain

( ∫_{B_{r}(x)} |[a, T]f(z)|^{p} dz)^{1/p} ≤ ∑_{k=0}^{∞} ( ∫_{B_{r}(x)} |[a, T]f_{k}(z)|^{p} dz)^{1/p} ≤

≤ c∥f_{0}∥_{L^{p}(ℝ^{n})} +

+c ∑_{k=1}^{∞} \{ ∫_{B_{r}(x)} ( ∫_{B_{2^{k+1}r}(x)\B_{2^{k}r}(x)} \frac{|a(y) − a(z)| |f(y)|}{|z − y|^{n}} dy)^{p} dz\}^{1/p} ≤

≤ cr^{λ/p}∥f∥_{L^{p,λ}(ℝ^{n})} +

+c ∑_{k=1}^{∞} \frac{1}{(2^{k}r)^{n}} \{ ∫_{B_{r}(x)} ( ∫_{B_{2^{k+1}r}(x)} |a(y) − a(z)| |f(y)| dy)^{p} dz\}^{1/p}.

For any σ > 0, we set

a_{σ} = \frac{1}{|B_{σ}(x)|} ∫_{B_{σ}(x)} a(y) dy.

By the John–Nirenberg’s lemma on BMO functions and the fact that |a_{2^{k+1}r} − a_{r}| ≤ c(n)(k + 1)∥a∥_{∗} (see [7]), we obtain

{ ∫_{B_{r}(x)} ( ∫_{B_{2^{k+1}r}(x)} |a(y) − a(z)| |f(y)| dy)^{p} dz}^{1/p} ≤

≤ { ∫_{B_{r}(x)} |a_{r} − a(z)|^{p} dz}^{1/p} ∫_{B_{2^{k+1}r}(x)} |f(y)| dy +

+cr^{n/p} ∫_{B_{2^{k+1}r}(x)} |a(y) − a_{r}| |f(y)| dy ≤

≤ cr^{n+λ/p}2^{k{n(1−1/p)+λ/p}}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})} +

+cr^{n/p}( ∫_{B_{2^{k+1}r}(x)} |a(y) − a_{r}|^{p′} dy)^{1/p′}( ∫_{B_{2^{k+1}r}(x)} |f(y)|^{p} dy)^{1/p} ≤

≤ cr^{n+λ/p}2^{k{n(1−1/p)+λ/p}}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})}cr^{n+λ/p}2^{k{n(1−1/p)+λ/p}} ×

×\{(\frac{1}{|B_{2^{k+1}r}(x)|} ∫_{B_{2^{k+1}r}(x)} |a(y) − a_{2^{k+1}r}|^{p′} dy)^{1/p′} +

+|a_{2^{k+1}r} − a_{r}|}∥f∥_{L^{p,λ}(ℝ^{n})} ≤

≤ c(k + 1)r^{n+λ/p}2^{k{n(1−1/p)+λ/p}}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})},

where, and in what follows, 1/p + 1/p′ = 1. Thus,

( ∫_{B_{r}(x)} |[a, T]f(z)|^{p} dz)^{1/p} ≤

≤ cr^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})}\{1 + ∑_{k=1}^{∞} \frac{k + 1}{2^{k(n−λ)/p}} \} ≤ cr^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})}.

Therefore

∥[a, T]f∥_{L^{p,λ}(ℝ^{n})} ≤ c∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})}.

From this theorem and Theorems 2.7 and 2.10 in [2], we easily deduce the following corollary.

Corollary 2.3. Let p ∈ (1, ∞), λ ∈ (0, n), and a ∈ BMO(ℝ^{n}). If k is a constant or a variable C − Z kernel on ℝ^{n} and T the corresponding operator, then there exists a constant c = c(n, p, λ, k) or c = c(n, p, λ, k, M) such that for all f ∈ L^{p,λ}(ℝ^{n}),

∥[a, T]f∥_{L^{p,λ}(ℝ^{n})} ≤ c∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ^{n})}.

From this and the extension theorem of BMO(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 of ℝ^{n} and a ∈ BMO(ℝ^{n}). 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 ∈ L^{p,λ}(Ω),

∥[a, T]f∥_{L^{p,λ}(Ω)} ≤ c∥a∥_{∗}∥f∥_{L^{p,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 Ω be an open subset of ℝ^{n}, a ∈ V MO(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 ρ_{0} = ρ_{0}(ε, η) such that for any ball B_{r} with the radius r ∈ (0, ρ_{0}), B_{r} ∩ Ω ≡ Ω_{r} ≠ ∅ and all f ∈ L^{p,λ}(Ω_{r}),

∥[a, T]f∥_{L^{p,λ}(Ω_{r})} ≤ cε∥f∥_{L^{p,λ}(Ω_{r})},

where c = c(a, p, λ, k) is independent of f and ε.

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 ℝ_{+}^{n} = {x = (x′, x_{n}) : x′ = (x_{1}, …, x_{n−1}) ∈ ℝ^{n−1}, x_{n} > 0}. To establish the boundary estimates of the solutions to (1.1), we need to study the boundedness on L^{p,λ}(ℝ_{+}^{n}) 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 x̃ = (x′, −x_{n}) for x = (x′, x_{n}) ∈ ℝ_{+}^{n}. If a sublinear operator T is bounded on L^{p}(ℝ_{+}^{n}) and for any f ∈ L^{1}(ℝ_{+}^{n}) with compact support and x ∈ ℝ_{+}^{n},

|T f(x)| ≤ c ∫_{ℝ_{+}^{n}} \frac{|f(y)|}{|x̃ − y|^{n}} dy,  (2.4)

then T is also bounded on L^{p,λ}(ℝ_{+}^{n}).

Proof. Let z ∈ ℝ_{+}^{n} and σ > 0. In what follows, we set B_{σ}^{+}(z) = B_{σ}(z) ∩ ℝ_{+}^{n}. We consider two cases.

Case 1. 0 ≤ z_{n} < 2σ. In this case, we write

f(y) = f(y)χ_{B_{2^{4}σ}^{+}(z)}(y) + ∑_{ℓ=4}^{∞} f(y)χ_{B_{2^{ℓ+1}σ}^{+}(z)\B_{2^{ℓ}σ}^{+}(z)}(y) ≡ ∑_{ℓ=3}^{∞} f_{ℓ}(y). (2.5)

Therefore, by L^{p}-boundedness of T and (2.4), we obtain

( ∫_{B_{σ}^{+}(z)} |T f(x)|^{p} dx)^{1/p} ≤ ∑_{ℓ=3}^{∞} ( ∫_{B_{σ}^{+}(z)} |T f_{ℓ}(x)|^{p} dx)^{1/p} ≤

≤ c∥f_{3}∥_{L^{p}(ℝ_{+}^{n})} + c ∑_{ℓ=4}^{∞} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+1}σ}^{+}(z)\B_{2^{ℓ}σ}^{+}(z)} \frac{|f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} + c ∑_{ℓ=4}^{∞} \frac{1}{(2^{ℓ}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+1}σ}^{+}(z)} |f(y)| dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}\{1 + ∑_{ℓ=4}^{∞} \frac{1}{2^{ℓ(n−λ)/p}} \} ≤ cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})},

which is the desirable estimate.

Case 2. There exists ℓ ∈ ℕ such that 2^{ℓ}σ ≤ z_{n} < 2^{ℓ+1}σ. In this case, we write

f(y) = f(y)χ_{B_{2^{ℓ+4}σ}^{+}(z)}(y) + ∑_{k=1}^{∞} f(y)χ_{B_{2^{ℓ+k+4}σ}^{+}(z)\B_{2^{ℓ+k+3}σ}^{+}(z)}(y) ≡

≡ ∑_{k=0}^{∞} f_{k}(y).  (2.6)

From (2.4), it follows that

( ∫_{B_{σ}^{+}(z)} |T f(x)|^{p} dx)^{1/p} ≤ c\{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+4}σ}^{+}(z)} \frac{|f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} +

+c ∑_{k=1}^{∞} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+k+4}σ}^{+}(z)\B_{2^{ℓ+k+3}σ}^{+}(z)} \frac{|f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} ≤

≤ \frac{c}{(2^{ℓ}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+4}σ}^{+}(z)} |f(y)| dy)^{p} dx\}^{1/p} +

+c ∑_{k=1}^{∞} \frac{1}{(2^{ℓ+k}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+k+4}σ}^{+}(z)} |f(y)| dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}\{\frac{1}{2^{ℓ(n−λ)/p}} + ∑_{k=1}^{∞} \frac{1}{2^{(ℓ+k)(n−λ)/p}} \} ≤ cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})},

by noting that ℓ ∈ 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) = {a_{in}(x)}_{i=1}^{n} be as in (1.2) and define

T (x, y) ≡ x − \frac{2x_{n}}{a_{nn}(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 ℝ_{+}^{n}. If k is a variable C − Z kernel on Ω, and for x ∈ Ω we define

T̃f(x) = ∫_{Ω} k(x, T (x) − y)f(y) dy

with T (x) = T (x, x), then there exists a constant c = c(p, λ, ν, k) such that for all f ∈ L^{p,λ}(Ω),

∥T̃f∥_{L^{p,λ}(Ω)} ≤ c∥f∥_{L^{p,λ}(Ω)}.

For the linear commutator on ℝ_{+}^{n}, we have

Theorem 2.4. Let p ∈ (1, ∞), λ ∈ (0, n), and a ∈ BMO(ℝ_{+}^{n}). If a linear operator T̃ satisfies (2.4) and [a, T̃] is bounded on L^{p}(ℝ_{+}^{n}), then [a, T̃] is also bounded on L^{p,λ}(ℝ_{+}^{n}).

Proof. Let z ∈ ℝ_{+}^{n} and σ > 0. Similarly to the proof of Theorem 2.3, we also consider two cases.

Case 1. 0 ≤ z_{n} < 2σ. In this case, we write f as in (2.5). We then have

( ∫_{B_{σ}^{+}(z)} |[a, T̃]f(x)|^{p} dx)^{1/p} ≤ ∑_{j=3}^{∞} ( ∫_{B_{σ}^{+}(z)} |[a, T̃]f_{j}(x)|^{p} dx)^{1/p} ≤

≤ c∥a∥_{∗}∥f_{3}∥_{L^{p}(ℝ_{+}^{n})} +

+c ∑_{j=4}^{∞} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{j+1}σ}^{+}(z)\B_{2^{j} σ}^{+}(z)} \frac{|a(y) − a(x)||f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} +

+c ∑_{j=4}^{∞} \frac{1}{(2^{j}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{j+1}σ}^{+}(z)} |a(y) − a(x)| |f(y)| dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} + c ∑_{j=4}^{∞} \frac{1}{(2^{j}σ)^{n}} ( ∫_{B_{σ}^{+}(z)} |a_{σ} − a(x)|^{p} dx)^{1/p} ×

×( ∫_{B_{2^{j+1}σ}^{+}(z)} |f(y)| dy) +

+c ∑_{j=4}^{∞} \frac{1}{(2^{j}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{j+1}σ}^{+}(z)}|a(y) − a_{σ}| |f(y)| dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} + cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}( ∑_{j=4}^{∞} \frac{1}{2^{j(n−λ)/p}} ) +

+c ∑_{j=4}^{∞} \frac{1}{(2^{j}σ)^{n}} (2^{j}σ)^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}σ^{n/p}( ∫_{B_{2^{j+1}σ}^{+}(z)} |a(y) − a_{σ}|^{p′} dy)^{1/p′} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} + cσ^{n/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}\{ ∑_{j=4}^{∞} \frac{1}{(2^{j}σ)^{(n−λ)/p}} (2^{j}σ)^{n/p′} ×

×[(\frac{1}{|B_{2^{j+1}σ}^{+}(z)|} ∫_{B_{2^{j+1}σ}^{+}(z)} |a(y) − a_{2^{j+1}σ}|^{p′} dy)^{1/p′} + |a_{2^{j+1}σ} − a_{σ}|]\} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} +

+cσ^{λ/p}∥f∥_{L^{p,λ}(ℝ_{+}^{n})} ∑_{j=4}^{∞} \frac{1}{2^{j(n−λ)/p}} \{∥a∥_{∗} + |a_{2^{j+1}σ} − a_{σ}|\} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}\{ ∑_{j=4}^{∞} \frac{j + 1}{2^{j(n−λ)/p}} \} ≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})},

where for any r > 0 we set

a_{r} = \frac{1}{|B_{r}^{+}(x)|} ∫_{B_{r}^{+}(x)} a(y) dy

and use the fact that λ < n and |a_{2^{j+1}σ} − a_{σ}| ≤ c(n)j∥a∥_{∗}. This estimation is the expected one.

Case 2. There exists ℓ ∈ ℕ such that 2^{ℓ}σ ≤ z_{n} < 2^{ℓ+1}σ. In this case, we write f as in (2.6). We then have

( ∫_{B_{σ}^{+}(z)} |[a, T̃]f(x)|^{p} dx)^{1/p} ≤ ∑_{k=0}^{∞} ( ∫_{B_{σ}^{+}(z)} |[a, T̃]f_{k}(x)|^{p} dx)^{1/p} ≤

≤ c\{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+4}σ}^{+}(z)} \frac{|a(y) − a(x)||f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} +

+c ∑_{k=1}^{∞} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{k+ℓ+4}σ}^{+}(z)\B_{2^{k+ℓ+3}σ}^{+}(z)} \frac{|a(y) − a(x)||f(y)|}{|x̃ − y|^{n}} dy)^{p} dx\}^{1/p} ≤

≤ \frac{c}{(2^{ℓ}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{ℓ+4}σ}^{+}(z)} |a(y) − a(x)||f(y)| dy)^{p} dx\}^{1/p} +

+c ∑_{k=1}^{∞} \frac{1}{(2^{k+ℓ}σ)^{n}} \{ ∫_{B_{σ}^{+}(z)} ( ∫_{B_{2^{k+ℓ+4}σ}^{+}(z)} |a(y) − a(x)| |f(y)| dy)^{p} dx\}^{1/p} ≤

≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})}\{ ∑_{k=1}^{∞} \frac{1}{2^{(k+ℓ)(n−λ)/p}} \} ≤ cσ^{λ/p}∥a∥_{∗}∥f∥_{L^{p,λ}(ℝ_{+}^{n})},

since ℓ ∈ 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 any σ > 0, B_{σ}^{+}(x) = {(x′, x_{n}) ∈ ℝ^{n} : |x| < σ, x_{n} > 0}. Let a ∈ V MO(ℝ_{+}^{n}) and η be its V MO modulus. If k is a variable C − Z kernel on ℝ_{+}^{n} and T̃ is as in Corollary 2.6, then for any ε > 0, there exists a positive ρ_{0} = ρ_{0}(ε, η) such that for any r ∈ (0, ρ_{0}) and f ∈ L^{p,λ}(B_{r}^{+}),

∥[a, T̃]f∥_{L^{p,λ}(B_{r}^{+})} ≤ cε∥f∥_{L^{p,λ}(B_{r}^{+})}

with c = c(ν, p, λ, k) independent of f, ε, and r.

3. Elliptic Equations with V MO Coefficients

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

Definition 3.1. Let p ∈ (1, ∞), λ ∈ (0, n), and Ω be an open subset of ℝ^{n}. f ∈ L_{loc}^{1}(Ω) is said to belong to the Sobolev–Morrey space W^{2}L^{p,λ}(Ω) if and only if u and its distributional derivatives, u_{x_{i}}, u_{x_{i}x_{j}} (i, j = 1, …, n) are in L^{p,λ}(Ω). Moreover, let ∥u∥_{W^{2}L^{p,λ}(Ω)} ≡ ∥u∥_{L^{p,λ}(Ω)}+∑_{i=1}^{n} ∥u_{x_{i}}∥_{L^{p,λ}(Ω)} + ∑_{i,j=1}^{n} ∥u_{x_{i}x_{j}}∥_{L^{p,λ}(Ω)}.

We also assume that f ∈ W_{loc}^{2}L^{p,λ}(Ω) if f ∈ W^{2}L^{p,λ}(Ω′) for every Ω′ ⊂⊂ Ω.

Now, let Ω be an open bounded subset of ℝ^{n} with n ≥ 3 and ∂Ω ∈ C^{1,1},

L ≡ ∑_{i,j=1}^{n} a_{ij}(x)\frac{∂^{2}}{∂x_{i}∂x_{j}}

with a_{ij}’s satisfying (1.2). We also assume that a_{ij} ∈ V MO(Ω). Since for each function in V MO(Ω) there is an extension to ℝ^{n} with the V MO modulus controlled by its original one (see [8, page 42 and 54]), without loss of generality, we may assume that a_{ij}’s belong to V MO(ℝ^{n}). Let f ∈ L^{p,λ}(Ω), p ∈ (1, ∞) and λ ∈ (0, n). We are interested in the following Cauchy problem:

\begin{cases}Lu = f a.e. in Ω, \\ u ∈ W^{2}L^{p,λ}(Ω) ∩ W_{0}^{1,p}(Ω). \end{cases}  (3.1)

Note that Ω is bounded; therefore f ∈ L^{p,λ}(Ω) implies f ∈ L^{p}(Ω). By the results in [3], we know that for f ∈ L^{p,λ}(Ω) with p ∈ (1, ∞) and λ ∈ (0, n), (3.1) has a unique solution u ∈ W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω) satisfying

∥u∥_{W^{2,p}(Ω)} ≤ c∥f∥_{L^{p}(Ω)},  (3.2)

where c is independent of f. Our main interest here is to improve (3.2) into

∥u∥_{W^{2}L^{p,λ}(Ω)} ≤ c∥f∥_{L^{p,λ}(Ω)}.  (3.3)

By a standard procedure, the proof of (3.3) consists in establishing the interior and boundary estimates of the solution to (3.1); see Theorems 4.4, 4.1, and 4.2 in [3]. Indeed, by a similar proof to Theorem 4.2 in [2] and Theorem 3.3 in [5], we can prove the following interior estimate.

Theorem 3.1. Let L satisfy the above assumption and η =(∑_{i,j=1}^{n} η_{ij}^{2})^{1/2} where η_{ij} is the V MO modulus of a_{ij} in Ω. Let λ ∈ (0, n), q, p ∈ (1, ∞), q ≤ p, f ∈ L^{p,λ}(Ω), u ∈ W^{2}L^{q,λ}(Ω) ∩ W_{0}^{1,q}(Ω), and Lu = f a.e. in Ω. Then u ∈ W_{loc}^{2}L^{p,λ}(Ω). Moreover, given any Ω′ ⊂⊂ Ω, there exists a constant c = c(n, p, λ, ν, dist(Ω′, ∂Ω), η) such that

∥u∥_{W^{2}L^{p,λ}(Ω′)} ≤ c{∥u∥_{L^{p,λ}(Ω)} + ∥f∥_{L^{p,λ}(Ω)}}.

The case q = p of 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 ∈ L^{p,λ}(Ω), u ∈ W^{2}L^{q,λ}(Ω) ∩ W_{0}^{1,q}(Ω), and Lu = f a.e. in Ω. Then u ∈ W^{2}L^{p,λ}(Ω) and there exists a constant c = c(n, p, λ, ν, ∂Ω, η) such that

∥u∥_{W^{2}L^{p,λ}(Ω)} ≤ c{∥u∥_{L^{p,λ}(Ω)} + ∥f∥_{L^{p,λ}(Ω)}}.

To prove Theorem 3.2, we need to introduce more notation. We let W_{γ_{0}}^{2,p}(B_{σ}^{+}) be the closure in W^{2,p} of the subspace

C_{γ_{0}} = {u : u is the restriction to B_{σ}^{+}
of a function in C_{0}^{∞}(B_{σ}^{+}) and u(x′, 0) = 0},

where, and in what follows, B_{σ} = {x ∈ ℝ^{n} : |x| < σ} and B_{σ}^{+} = {(x′, x_{n}) ∈ ℝ^{n} : |x| < σ and x_{n} > 0} for any σ > 0. We also make the following assumption and refer to it as assumption (H).

\begin{cases}Let n ≥ 3, b_{ij} ∈ V MO(ℝ^{n}), i, j = 1, …, n, \\ b_{ij}(x) = b_{ji}(x), i, j = 1, …, n, a.e. in B_{σ}^{+}. \\ There exists μ > 0 such that for all ξ ∈ ℝ^{n}, \\ μ^{−1}|ξ|^{2} ≤ ∑_{i,j=1}^{n} b_{ij}(x)ξ_{i}ξ_{j} ≤ μ|ξ|^{2}, a.e. in B_{σ}^{+}. \end{cases}  (H)

Let

L̃ ≡ ∑_{i,j=1}^{n} b_{ij} \frac{∂^{2}}{∂x_{i}∂x_{j}}

and

Γ(x, t) = \frac{1}{(n − 2)ω_{n}(detb_{ij})^{1/2}} ( ∑_{i,j=1}^{n} B_{ij}(x)t_{i}t_{j})^{(2−n)/2},

Γ_{i}(x, t) = \frac{∂}{∂t_{i}} Γ(x, t), Γ_{ij}(x, t) = \frac{∂^{2}}{∂t_{i}∂t_{j}} Γ(x, t),

for a.e. x ∈ B_{σ}^{+} and all t ∈ ℝ^{n} \ {0}, where B_{ij}’s are the entries of the inverse of the matrix {b_{ij}}_{i,j=1,...,n}. We also set b(x) = {b_{in}(x)}_{i=1}^{n} and

T (x, y) = x − \frac{2x_{n}}{b_{nn}(y)}b(y).

By a covering and flattening argument, the proof of Theorem 3.2 can be reduced to establishing the estimate in L^{p,λ}(B_{r}^{+}) of u_{x_{i}x_{j}} with u ∈ W_{γ_{0}}^{2,q}(B_{r}^{+}) and L̃u ∈ L^{p,λ}(B_{r}^{+}), 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 let u ∈ W_{γ_{0}}^{2,p}(B_{σ}^{+}) with p ∈ (1, ∞). Then

u_{x_{i}x_{j}}(x) =

= p.v. ∫_{B_{σ}^{+}} Γ_{ij}(x, x − y){ ∑_{h,k=1}^{n}[b_{hk}(x) − b_{hk}(y)]u_{x_{h}x_{k}}(y) + L̃u(y)} dy

+L̃u(x) ∫_{|t|=1} Γ_{i}(x, y)t_{j} dσ_{t} + I_{ij}(x),  (3.4)

where for i, j = 1, …, n − 1,

I_{ij}(x) =

= p.v. ∫_{B_{σ}^{+}} Γ_{ij}(x, T (x) − y){ ∑_{h,k=1}^{n}[b_{hk}(x) − b_{hk}(y)]u_{x_{h}x_{k}}(y) + L̃u(y)} dy;

for i = 1, …, n − 1,

I_{in}(x) = I_{ni}(x) = ∫_{B_{σ}^{+}} ( ∑_{ℓ=1}^{n} B_{ℓ}(x)Γ_{iℓ}(x, T (x) − y)){⋯ } dy,

and

I_{nn}(x) = ∫_{B_{σ}^{+}} ∑_{ℓ,k=1}^{n} B_{ℓ}(x)B_{k}(x)Γ_{ℓk}(x, T (x) − y){⋯ } dy;

in the formulas above T (x) = T (x; x), B_{i}(x) is the i-th component of the vector B(x) = T (e_{n}; x) with e_{n} = (0, …, 0, 1), t_{j} 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.

Now, to finish the proof of Theorem 3.2, we only need to prove the following theorem.

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

Set η̃ = ( ∑_{i,j=1}^{n} η_{ij}^{2})^{1/2}, where η̃_{ij} is the V MO modulus of b_{ij}, and

M ≡ max_{i,j=1,...,n} max_{|α|≤2n} ∥\frac{∂^{α}}{∂t^{α}} Γ_{ij}(x, t)∥_{L^{∞}(B_{σ}^{+}×Σ)}.

Then there exists a positive number ρ_{0} = ρ_{0}(n, q, p, M, μ, η̃, λ), ρ_{0} < σ, such that for any r ∈ (0, ρ_{0}) and any u ∈ W_{γ_{0}}^{2,q}(B_{r}^{+}) with L̃u ∈ L^{p,λ}(B_{r}^{+}), we have u ∈ W^{2}L^{p,λ}(B_{r}^{+}). Furthermore, there exists a constant c=c(n, p, λ, M, μ, η̃) such that

∥u_{x_{i}x_{j}}∥_{L^{p,λ}(B_{r}^{+})} ≤ c∥L̃u∥_{L^{p,λ}(B_{r}^{+})}.  (3.5)

Proof. Set for i, j, h, k = 1, …, n

S_{ijhk}(f)(x) = p.v. ∫_{B_{r}^{+}} Γ_{ij}(x, x − y)(b_{hk}(x) − b_{hk}(y))f(y) dy,

and for i, j = 1, …, n − 1, h, k = 1, …, n

S̃_{ijhk}(f)(x) = ∫_{B_{r}^{+}} Γ_{ij}(x, T (x) − y)(b_{hk}(x) − b_{hk}(y))f(y) dy,

for i = 1, …, n − 1, h, k = 1, …, n

S̃_{inhk}(f)(x) = ∫_{B_{r}^{+}} ( ∑_{j=1}^{n} Γ_{ij}(x, T (x) − y)B_{j}(x))(b_{hk}(x) − b_{hk}(y))f(y) dy,

and, finally, for h, k = 1, …, n

S̃_{nnhk}(f)(x) =

= ∫_{B_{r}^{+}} ( ∑_{i,j=1}^{n} Γ_{ij}(x, T (x) − y)B_{i}(x)B_{j}(x))(b_{hk}(x) − b_{hk}(y))f(y) dy,

where r ∈ (0, σ] and f ∈ L^{s,λ}(B_{r}^{+}).

By Lemma 3.1 in [2] and Corollaries 2.5 and 2.7, we can fix ρ_{0} so small that

∑_{i,j,h,k} ∥S_{ijhk} + S̃_{ijhk}∥ < 1,

where the norm of operators S_{ijhk} + S̃_{ijhk} is the norm in the space of linear operators from L^{s,λ}(B_{r}^{+}) into itself if r ∈ (0, ρ_{0}) and s ∈ [q, p].

Consider u ∈ W_{γ_{0}}^{2,p}(B_{r}^{+}) with L̃u ∈ L^{p,λ}(B_{r}^{+}), r ∈ (0, ρ_{0}) and set

h_{ij} = p.v. ∫_{B_{r}^{+}} Γ_{ij}(x, x − y)L̃u(y) dy + L̃u(x) ∫_{|t|=1} Γ_{i}(x, t)t_{j} dσ_{t} + Ĩ_{ij}(x),

where

Ĩ_{ij}(x) = \{ \begin{matrix}∫_{B_{r}^{+}} Γ_{ij}(x, T (x) − y)L̃u(y) dy for i, j = 1, …, n − 1, \\ ∫_{B_{r}^{+}} ( ∑_{ℓ=1}^{n} Γ_{iℓ}(x, T (x) − y)B_{ℓ}(x))L̃u(y) dy \\ for i = 1, …, n − 1, j = n, \\ ∫_{B_{r}^{+}} ( ∑_{ℓ,m=1}^{n} Γ_{ℓm}(x, T (x)−y)B_{ℓ}(x)B_{m}(x))L̃u(y) dy for i=j =n. \end{matrix}

From Corollary 2.6, we easily deduce that h_{ij} ∈ L^{p,λ}(B_{r}^{+}).

Consider w ∈ [L^{p,λ}(B_{r}^{+})]^{n^{2}} and define T w : [L^{p,λ}(B_{r}^{+})]^{n^{2}} → [L^{p,λ}(B_{r}^{+})]^{n^{2}} by setting

T w = ((T w)_{ij})_{i,j=1,...,n} = ( ∑_{h,k=1}^{n}(S_{ijhk} + S̃_{ijhk})(w_{ij}) + h_{ij})_{i,j=1,...,n}.

The operator T is a contraction on [L^{p,λ}(B_{r}^{+})]^{n^{2}} and thus has a unique fixed point w̃. Since, by (3.4), {u_{x_{i}x_{j}}}_{i,j=1,...,n} is also a fixed point in [L^{q,λ}(B_{r}^{+})]^{n^{2}} and L^{p,λ}(B_{r}^{+}) ⊆ L^{q,λ}(B_{r}^{+}) if q ≤ p, the uniqueness of the fixed point implies that u_{x_{i}x_{j}} = w̃_{ij} ∈ L^{p,λ}(B_{r}^{+}) for i, 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

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

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(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 The People’s Republic of China