Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 271–277

LINDELÖF THEOREMS FOR MONOTONE SOBOLEV FUNCTIONS

Toshihide Futamura and Yoshihiro Mizuta

Hiroshima University, Graduate School of Science, Department of Mathematics Higashi-Hiroshima 739-8526, Japan; toshi@mis.hiroshima-u.ac.jp Hiroshima University, Faculty of Integrated Arts and Sciences The Division of Mathematical and Information Sciences Higashi-Hiroshima 739-8521, Japan; mizuta@mis.hiroshima-u.ac.jp

Abstract. This paper deals with Lindelöf type theorems for monotone functions in weighted Sobolev spaces.

1. Introduction

Let R^{n}, n ≥ 2, denote the n-dimensional Euclidean space. We use the notation D to denote the upper half space of R^{n}, that is,

D = {x = (x_{1}, …, x_{n−1}, x_{n}) : x_{n} > 0}.

Denote by B(x, r) the open ball centered at x with radius r, and set σB(x, r) = B(x, σr) for σ > 0 and S(x, r) = ∂B(x, r).

A continuous function u on D is called monotone in the sense of Lebesgue (see [5]) if for every relatively compact open set G ⊂ D,

max_{\overline{G}} u = max_{∂G} u  and  min_{\overline{G}} u = min_{∂G} u.

If u is monotone in D and p > n − 1, then

(1)  |u(x) − u(x′)| ≤ Mr(\frac{1}{r^{n}} ∫_{B(y,2r)} |∇u(z)|^{p} dz)^{1/p}

for every x, x′ ∈ B(y, r), whenever B(y, 2r) ⊂ D (see [6, Theorem 1] and [4, Theorem 2.8]). For further results of monotone functions, we refer to [3], [13] and [15].

Our aim in the present note is to extend the second author’s result [12, Theorem 2] and the most recent results by Manfredi–Villamor [8].

2000 Mathematics Subject Classification: Primary 31B25, 46E35.

Theorem 1. Let u be a monotone function on D satisfying

(2)  ∫_{D} |∇u(z)|^{p}z_{n}^{α} dz < ∞,

where p > n − 1 and 0 ≤ n + α − p < 1. Define

E_{n+α−p} = {ξ ∈ ∂D : lim sup_{r→0} r^{p−α−n} ∫_{B(ξ,r)∩D} |∇u(z)|^{p}z_{n}^{α} dz > 0}.

If ξ ∈ ∂D − E_{n+α−p} and there exists a curve γ in D tending to ξ along which u has a finite limit, then u has a nontangential limit at ξ.

Remark 1. We know that E_{n+α−p} has (n + α − p)-dimensional Hausdorff measure zero, and hence it is of C_{1−α/p,p} -capacity zero; for these results, see Meyers [9], [10] and the second author’s book [13].

We shall give a generalization of Theorem 1 (see Theorem 2 below). We proceed to the proof of Theorem 1 for the sake of clarity.

Throughout this paper, let M denote various positive constants independent of the variables in question, and M(ε) a positive constant which depends on ε.

2. Proof of Theorem 1

A sequence {X_{j}} is called regular at ξ ∈ ∂D if X_{j} → ξ and

|X_{j} − ξ| < c|X_{j+1} − ξ|

for some constant c > 0.

First we give the following result, which can be proved by (1).

Lemma 1. Let u be a monotone function on D satisfying (2) with n − 1 < p ≤ α + n. If ξ ∈ ∂D − E_{n+α−p} and there exists a regular sequence {X_{j}} ⊂ D with X_{j} = ξ + (0, …, 0, r_{j}) such that u(X_{j}) has a finite limit, then u has a nontangential limit at ξ.

Proof of Theorem 1. For r > 0 sufficiently small, take C(r) ∈ γ ∩ S(ξ, r). Letting C_{1}(r) = ξ + (0, …, 0, r), take an end point C_{2}(r) ∈ ∂D of a quarter of circle containing C_{1}(r) and C(r).

Let ϱ_{D}(x) denote the distance of x ∈ D from the boundary ∂D, that is, ϱ_{D}(x) = x_{n}. We take a finite covering {B(X_{j}, 4^{−1}ϱ_{D}(X_{j}))} of circular arc

C(r)C_{1}(r) such that

(i) X_{1} = C(r) and X_{N+1} = C_{1}(r); (ii) |z − ξ| < 2r and |z − C_{2}(r)| ∼ ϱ_{D}(z) for z ∈ A(ξ, r) = ⋃_{j} 2B_{j}, where

B_{j} = B(X_{j}, 4^{−1}ϱ_{D}(X_{j}));

Lindelöf theorems for monotone Sobolev functions 273

(iii) B_{j} ∩ B_{j+1} ≠ ∅ for each j; (iv) ∑_{j} χ_{2B_{j}} is bounded, where χ_{A} denotes the characteristic function of A; see Heinonen [2] and Hajłasz–Koskela [1]. By the monotonicity of u we see that

|u(x) − u(X_{j})| ≤ Mϱ_{D}(X_{j})(\frac{1}{ϱ_{D}(X_{j})^{n}} ∫_{2B_{j}} |∇u(z)|^{p} dz)^{1/p}

for x ∈ B_{j}. We have by Hölder’s inequality

|u(C_{1}(r)) − u(C(r))| ≤ |u(X_{1}) − u(X_{2})| + |u(X_{2}) − u(X_{3})|

+ ⋯ + |u(X_{N}) − u(X_{N+1})|

≤ M ∑_{j} ϱ_{D}(X_{j})^{1−(n−δ)/p}(∫_{2B_{j}} |∇u(z)|^{p}ϱ_{D}(X_{j})^{−δ} dz)^{1/p}

≤ M(∑_{j} ϱ_{D}(X_{j})^{p′(p−n+δ)/p})^{1/p′}

× (∫_{A(ξ,r)} |∇u(z)|^{p}ϱ_{D}(z)^{−δ} dz)^{1/p}

≤ M(∑_{j} ϱ_{D}(X_{j})^{p′(p−n+δ)/p})^{1/p′}

× (∫_{B(ξ,2r)∩D} |∇u(z)|^{p}ϱ_{D}(z)^{α}|C_{2}(r) − z|^{−δ−α} dz)^{1/p}

for δ > 0, where 1/p + 1/p′ = 1. Here note that

∑_{j} ϱ_{D}(X_{j})^{p′(p−n+δ)/p} ≤ M ∫_{A(ξ,r)} ϱ_{D}(z)^{p′(p−n+δ)/p−n} dz
≤ M ∫_{A(ξ,r)} |C_{2}(r) − z|^{p′(p−n+δ)/p−n} dz
≤ Mr^{p′(p−n+δ)/p}

when δ > n − p. Moreover,

(3) ∫_{2^{−j}}^{2^{−j+1}} |C_{2}(r) − z|^{−δ−α} dr ≤ ∫_{2^{−j}}^{2^{−j+1}} |r − |z||^{−δ−α} dr ≤ M2^{−j(1−δ−α)}

when −α < δ < 1 − α. Hence it follows that ∫_{2^{−j}}^{2^{−j+1}} |u(C_{1}(r))−u(C(r))|^{p} dr/r ≤ M2^{−j(p−n−α)} ∫_{B(ξ,2^{−j+2})∩D}|∇u(z)|^{p}ϱ_{D}(z)^{α} dz.

Since ξ ∈ ∂D − E_{n+α−p}, we can find a sequence {r_{j}} such that 2^{−j} < r_{j} < 2^{−j+1} and

lim_{j→∞}|u(C_{1}(r_{j})) − u(C(r_{j}))| = 0.

By our assumption we see that u(C_{1}(r_{j})) has a finite limit as j → ∞. If we note that {C_{1}(r_{j})} is regular at ξ, then Lemma 1 proves the required conclusion of the theorem.

3. Monotone functions on a measure space (D; μ)

Let μ be a Borel measure on R^{n} satisfying the doubling condition:

μ(2B) ≤ Mμ(B)

for every ball B ⊂ R^{n}. We further assume that

(4)  \frac{μ(B′)}{μ(B)} ≥ M(\frac{diam(B′)}{diam(B)})^{s}

for all B′ = B(ξ′, r′) and B = B(ξ, r) with ξ′, ξ ∈ ∂D and B′ ⊂ B, where s > 1 and diam(B) denotes the diameter of B.

A pair (u, g) ∈ L_{loc}^{1}(D; μ)×L_{loc}^{p}(D; μ) is said to satisfy p-Poincaré inequality if g ≥ 0 on D and

\frac{1}{μ(B)} ∫_{B} |u(x) − u_{B}| dμ(x) ≤ M diam(B)(\frac{1}{μ(σB)} ∫_{σB} g(z)^{p} dμ(z))^{1/p}

for every ball B with σB ⊂ D, where σ > 1 and

u_{B} = ∫−_{B} u(y) dμ(y) = \frac{1}{μ(B)} ∫_{B} u(y) dμ(y).

We need a stronger property than Poincaré inequalities; a continuous function u is called monotone in D if there exists a nonnegative function g ∈ L_{loc}^{p}(D; μ) such that

(5)  |u(x) − u_{B}| ≤ Mr(\frac{1}{μ(σB)} ∫_{σB} g(z)^{p} dμ(z))^{1/p}

for every x ∈ B with σB ⊂ D, where σ > 1 and B = B(y, r).

Now we show the following result, which gives of course a generalization of Theorem 1.

Theorem 2. Let u be a monotone function on D with g satisfying (5) and

(6)  ∫_{D} g(z)^{p} dμ(z) < ∞.

Suppose p > s − 1, and set

E = {ξ ∈ ∂D : lim sup_{r→0}(r^{−p}μ(B(ξ, r)))^{−1} ∫_{B(ξ,r)∩D} g(z)^{p} dμ(z) > 0}.

If ξ ∈ ∂D − E and there exists a curve γ in D tending to ξ along which u has a finite limit, then u has a nontangential limit at ξ.

Remark 2. Let 1 ≤ q < p/(n − 1). Let w be a Muckenhoupt (A_{q}) weight, and define

dμ(y) = w(y) dy.

If u is monotone in the sense of Lebesgue, then (u, |∇u|) satisfies the monotonicity property (5) by applying Hölder’s inequality to (1) with p replaced by p/q (see also Manfredi–Villamor [8]). If in addition u satisfies (6) with g = |∇u|, then we apply Theorem 1 with p replaced by p/q to obtain the same conclusion as Theorem 2.

Remark 3. In Theorem 2, since μ(E) = 0, we see that E is of C_{1,p,μ} - capacity zero; here the weighted p-capacity C_{1,p,μ}(E) is defined by

C_{1,p,μ}(E) = inf{∫ |f(y)|^{p} dμ : ∫_{B(x,1)} |x − y|^{1−n}f(y) dy ≥ 1 for all x ∈ E },

which has the property

(7)  C_{1,p,μ}(B(x, r)) ≤ Mr^{−p}μ(B(x, r)).

For proofs of these facts, see Meyers [9] and [10].

Proof of Theorem 2. By the monotonicity of u we see that

|u(x) − u(C(r))| ≤ M diam(B)(\frac{1}{μ(σB)} ∫_{σB} g(z)^{p} dμ(z))^{1/p}

for x ∈ B = B(C(r), 2^{−1}σ^{−1}ϱ_{D}(C(r))). We take a finite covering {B_{j}} of circular

arc C(r)C_{1}(r) as in the proof of Theorem 1; in this case

B_{j} = B(X_{j}, 2^{−1}σ^{−1}ϱ_{D}(X_{j})).

276 Toshihide Futamura and Yoshihiro Mizuta We find by Hölder’s inequality

|u(C_{1}(r)) − u(C(r))| ≤ |u(X_{1}) − u(X_{2})| + |u(X_{2}) − u(X_{3})|

+ ⋯ + |u(X_{N}) − u(X_{N+1})|

≤ M ∑_{j} ϱ_{D}(X_{j})^{1+δ/p}μ(σB_{j})^{−1/p}

× (∫_{σB_{j}} g(z)^{p}ϱ_{D}(z)^{−δ} dμ(z))^{1/p}

≤ M(∑_{j} ϱ_{D}(X_{j})^{p′(1+δ/p)}μ(σB_{j})^{−p′/p})^{1/p′}

× (∫_{A(ξ,r)} g(z)^{p}ϱ_{D}(z)^{−δ} dμ(z))^{1/p}

≤ M(∑_{j} ϱ_{D}(X_{j})^{p′(1+δ/p)}μ(σB_{j})^{−p′/p})^{1/p′}

× (∫_{B(ξ,2r)∩D} g(z)^{p}|C_{2}(r) − z|^{−δ} dμ(z))^{1/p}

for δ > 0, where 1/p + 1/p′ = 1. If we take δ > s − p, then we see from (4) that

∑_{j} ϱ_{D}(X_{j})^{p′(p+δ)/p}μ(σB_{j})^{−p′/p} ≤ M ∫_{0}^{2r} t^{p′(p+δ)/p}μ(B(C_{2}(r), t))^{−p′/p} dt/t

≤ Mr^{p′s/p}μ(B(ξ, 4r))^{−p′/p} ∫_{0}^{2r} t^{p′(p+δ−s)/p} dt/t

≤ Mr^{p′δ/p}(r^{−p}μ(B(ξ, r)))^{−p′/p}.

Hence it follows from (3) with 0 < δ < 1 and α = 0 that

∫_{2^{−j}}^{2^{−j+1}}|u(C_{1}(r))−u(C(r))|^{p} dr/r ≤ M(2^{jp}μ(B(ξ, 2^{−j})))^{−1} ∫_{B(ξ,2^{−j+2})}g(z)^{p} dμ(z). Thus we can show that u has a nontangential limit at ξ, in the same manner as Theorem 1.

Remark 4. Let u be a monotone Sobolev function on D satisfying ∫_{D} |∇u(x)|^{p} dμ(x) < ∞.

Lindelöf theorems for monotone Sobolev functions 277

Define

E_{1} = {ξ ∈ ∂D : ∫_{B(ξ,1)∩D} |ξ − y|^{1−n}|∇u(y)| dy = ∞}

and

E_{2} = {ξ ∈ ∂D : lim sup_{r→0}(r^{−p}μ(B(ξ, r)))^{−1} ∫_{B(ξ,r)∩D} |∇u(y)|^{p} dμ(y) > 0}.

Then we can show as in [11], [12] that u has a nontangential limit at every ξ ∈ ∂D − (E_{1} ∪ E_{2}). Note here that E_{1} ∪ E_{2} is of C_{1,p,μ} -capacity zero.

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Received 18 February 2002