Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 61–88

EXTREMAL LENGTH OF VECTOR MEASURES

Hiroaki Aikawa and Makoto Ohtsuka

Shimane University, Department of Mathematics Matsue 690-8504, Japan; haikawa@riko.shimane-u.ac.jp 971-258 Mogusa, Hino-shi, Tokyo 191-0033, Japan

Abstract. We define the extremal length of vector measures. We show the reciprocal relation between extremal distance and extremal width of vector measures.

1. Introduction

The notion of extremal length and modulus of curve families has been studied extensively and gives a lot of applications to complex analysis and potential theory. In particular, the coincidence between modulus and p-capacity plays an important role. On the other hand, the interest on degenerate elliptic equations is increasing in these years (see e.g. [6]). As far as we know, however, there has been no such result for a modulus and a capacity associated with degenerate elliptic equations. The main aim of this paper is to define a modulus of vector measures and to establish a coincidence as before for a certain degenerate elliptic operator.

Throughout this paper we work with the Euclidean space R^{n} with n ≥ 2. Let 1 < p < ∞ and 1/p+1/p′ = 1. Let Ω be a domain in R^{n} and let 𝒜 = 𝒜 (x) be a positive definite symmetric n × n-matrix with measurable components a_{ij} such that

c_{0}^{−2}w(x)^{2/p}|ξ|^{2} ≤ ∑_{i,j} a_{ij}(x)ξ_{i}ξ_{j} ≤ c_{0}^{2}w(x)^{2/p}|ξ|^{2}

for any vector ξ = (ξ_{1}, …, ξ_{n}) ∈ R^{n} and x ∈ Ω, where c_{0} ≥ 1 is a constant and 0 < w < ∞ is a weight which indicates the degeneracy of 𝒜 . It is natural to define the capacity by

inf_{u∈𝒟} ∫_{Ω}(∑_{i,j} a_{ij} \frac{∂u}{∂x_{i}} \frac{∂u}{∂x_{j}} )^{p/2} dx,

where 𝒟 is a certain class of smooth functions, or more generally precise functions. If {a_{ij}} is the identity matrix, then the above capacity coincides with the classical p-capacity. We shall define a generalized modulus closely related to the above capacity. In particular, we shall generalize the classical extremal distance and width so that the reciprocal relationship between them remains in force.

1991 Mathematics Subject Classification: Primary 31B15.

The first author was supported in part by Japanese Ministry of Education, Science and Culture Grant-in-Aid for Scientific Research (B) (No. 09440062).

First we recall the definition of the modulus of a system of measures developed by Fuglede [4, Chapter I]. Let f be a nonnegative Borel measurable function and let μ be a nonnegative Borel measure. If ∫ f dμ ≥ 1, then we write f ∧ μ. Let ℰ be a system of nonnegative Borel measures. If f ∧ μ for all μ ∈ ℰ , then we write f ∧ ℰ . We define the modulus M_{p}(ℰ) of ℰ by

M_{p}(ℰ) = inf{∫ f^{p} dx : f ≥ 0, f ∧ ℰ },

where the infimum is understood to be infinity if there is no feasible f.

We generalize the above notions in connection with weights in the Muckenhoupt A_{p} class. Hereafter let w be a weight in the Muckenhoupt A_{p} class and let ∥f∥_{p,w} = (∫ f^{p}w dx)^{1/p}. Let L_{w}^{p}(Ω) be the space of all functions f on Ω such that ∥f∥_{p,w} < ∞. The A_{p} -condition will be required for many properties analogous to those for the unweighted case, e.g. approximation of f ∈ L_{w}^{p}(Ω) by a smooth function (see Propositions 3–5 in Section 2). See also Kilpeläinen [7]. We define the weighted modulus M_{p,w}(ℰ) by

M_{p,w}(ℰ) = inf{∫ f^{p}w dx : f ≥ 0, f ∧ ℰ },

where the infimum is understood to be infinity if there is no feasible f. Observe that if w = w′ a.e., then M_{p,w}(ℰ) = M_{p,w′}(ℰ). Hereafter, we assume that w is defined everywhere and 0 < w < ∞. Let w^{−1/p}ℰ = {w^{−1/p}μ : μ ∈ ℰ }. Then, by definition, M_{p,w}(ℰ) = M_{p}(w^{−1/p}ℰ) and M_{p}(ℰ) = M_{p,w}(w^{1/p}ℰ). In particular, M_{p,w}(ℰ) = 0 if and only if M_{p}(w^{−1/p}ℰ) = 0. Note that if w = w′ a.e., then the systems w^{−1/p}ℰ and w′^{−1/p}ℰ may differ, though M_{p}(w^{−1/p}ℰ) = M_{p,w}(ℰ) = M_{p,w′}(ℰ) = M_{p}(w′^{−1/p}ℰ). It is essential that w is defined everywhere and 0 < w < ∞. (If w could be ∞ on some set, then w^{−1/p}ℰ might include a zero measure and M_{p}(w^{−1/p}ℰ) = ∞.) If M_{p,w}(ℰ_{0}) = 0, then we say that ℰ_{0} is (p, w)-exceptional (abbreviated to (p, w)-exc.). If a statement concerning nonnegative Borel measures fails to hold only for (p, w)-exc. set of measures in ℰ_{0} , then we say that it holds (p, w)-a.e.

Fuglede [4, Chapter I] proved several fundamental properties of M_{p,w}. By |E| we denote the Lebesgue measure of E.

(i) M_{p,w}(ℰ) ≤ M_{p,w}(ℰ′) if ℰ ⊂ ℰ′. (ii) M_{p,w}(ℰ) ≤ ∑ M_{p,w}(ℰ_{j}) if ℰ = ⋃ ℰ_{j}. (iii) If \overline{μ} is the completion of μ and |E| = 0, then \overline{μ}(E) = 0 for (p, w)-a.e. μ. (iv) If f ∈ L_{w}^{p}(R^{n}), then f is \overline{μ}-integrable for (p, w)-a.e. μ.

(v) If ∥f_{i} − f∥_{p,w} → 0, there exists a subsequence f_{i_{j}} such that ∫ |f_{i_{j}} −f| d\overline{μ} → 0 for (p, w)-a.e. μ.

(vi) M_{p,w}(ℰ) = 0 if and only if there is f ≥ 0 with ∥f∥_{p,w} < ∞ such that

∫ f dμ = ∞ for all μ ∈ ℰ . (vii) There is f ≥ 0 such that f ∧ μ for (p, w)-a.e. μ ∈ ℰ and M_{p,w}(ℰ) = ∥f∥_{p,w}^{p}.

In view of the property (vii) we have

M_{p,w}(ℰ) = inf{∫ f^{p}w dx : f ≥ 0, f ∧ ℰ (p, w)-a.e.}.

Now we define the modulus of a system of vector measures. Let ν be a vector measure whose components ν_{i} are signed measures. The total variation |ν| of ν is defined by

|ν|(E) = sup ∑_{j}(∑_{i=1}^{n} ν_{i}(E_{j})^{2})^{1/2} for Borel sets E ,

where the supremum is taken over all finite partitions {E_{j}} of E into Borel sets. The total variation |ν| is a nonnegative measure. Let ξ = (ξ_{1}, …, ξ_{n}) be a vector valued function. If ∫ |ξ_{i}|d|ν_{i}| < ∞ for i = 1, …, n, then we define ∫ ξ · dν = ∑_{i=1}^{n} ∫ ξ_{i} dν_{i}. It is known that | ∫ ξ · dν| ≤ ∫ |ξ| d|ν|. See [9, Chapter 13] for details. We give the notion of exceptional sets of vector measures.

Definition. Let ℱ_{0} be a set of vector measures ν. We put |ℱ_{0}| = {|ν| : ν ∈ ℱ_{0}}. If M_{p,w}(|ℱ_{0}|) = 0, then we say that ℱ_{0} is (p, w)-exceptional (abbreviated to (p, w)-exc.). If a statement concerning vector measures ν fails to hold only for (p, w)-exc. system ℱ_{0} , then we say that it holds (p, w)-a.e.

Let 𝒜 = 𝒜 (x) be a positive definite symmetric n×n-matrix with measurable components a_{ij} such that

c_{0}^{−2}w(x)^{2/p}|ξ|^{2} ≤ ∑_{i,j} a_{ij}(x)ξ_{i}ξ_{j} ≤ c_{0}^{2}w(x)^{2/p}|ξ|^{2}

for any vector ξ = (ξ_{1}, …, ξ_{n}) ∈ R^{n} and x ∈ Ω, where c_{0} ≥ 1 is a constant. Let ℬ = ℬ(x) = {b_{ij}} be the inverse matrix of 𝒜 . It is easy to see that ℬ satisfies

c_{0}^{−2}w(x)^{−2/p}|ξ|^{2} ≤ ∑_{i,j} b_{ij}(x)ξ_{i}ξ_{j} ≤ c_{0}^{2}w(x)^{−2/p}|ξ|^{2}.

For simplicity we let 𝒜 [ξ] = (^{t}ξ𝒜 ξ)^{1/2} = (∑_{i,j} a_{ij}(x)ξ_{i}ξ_{j})^{1/2} for ξ ∈ R^{n}; and if ξ = ξ(x) is a vector valued measurable function on Ω, then we let 𝒜_{p}(ξ) = (∫_{Ω} 𝒜 [ξ]^{p} dx)^{1/p}. Similarly, ℬ[ξ] and ℬ_{p}(ξ) are defined. Observe that 𝒜 and ℬ can be written as 𝒜 = \sqrt{𝒜} ^{2} and ℬ = \sqrt{ℬ} ^{2} with positive symmetric matrices \sqrt{𝒜} and \sqrt{ℬ} . Then 𝒜 [ξ] = |\sqrt{𝒜} ξ|, ℬ[ξ] = |\sqrt{ℬ} ξ|, 𝒜_{p}(ξ) = (∫_{Ω} |\sqrt{𝒜} ξ|^{p} dx)^{1/p} and ℬ_{p}(ξ) =( ∫_{Ω} |\sqrt{ℬ} ξ|^{p} dx)^{1/p}. We remark that

(1.1)  \begin{aligned}c_{0}^{−1}w(x)^{1/p}|ξ| & ≤ |\sqrt{𝒜} ξ| ≤ c_{0}w(x)^{1/p}|ξ|, \\ c_{0}^{−1}w(x)^{−1/p}|ξ| & ≤ |\sqrt{ℬ} ξ| ≤ c_{0}w(x)^{−1/p}|ξ|. \end{aligned}

In the same way as in the remark on M_{p}(w^{−1/p}ℰ), we emphasize that the matrices 𝒜 and ℬ are defined everywhere and satisfy the above inequalities. If 𝒜 is not defined for some set of measure 0, then 𝒜 should be defined to be the identity matrix on that set.

Definition. Let ξ = (ξ_{1}, …, ξ_{n}) be a vector valued function and let ν = (ν_{1}, …, ν_{n}) be a vector measure defined on Ω. We write ξ ∧ ν if ∫ ξ · dν ≥ 1. Let ℱ be a set of complete vector measures. We write ξ ∧ ℱ (p, w)-a.e. if ξ ∧ ν for (p, w)-a.e. ν ∈ ℱ. We define

M_{𝒜 ,p} (ℱ) = inf{𝒜_{p}(ξ)^{p} : ξ ∧ ℱ (p, w)-a.e. },
M_{ℬ,p′} (ℱ) = inf{ℬ_{p′}(ξ)^{p′} : ξ ∧ ℱ (p′, w^{1−p′})-a.e. },

where the infima are understood to be infinity if there is no feasible ξ.

Remark. In general we cannot remove ‘(p, w)-a.e.’ and ‘(p′, w^{1−p′})-a.e.’ in the above definition. In fact, let us consider the line segment L between (0, …, 0) and (1, 0, …, 0). Let ℱ = {dx_{1}|_{L}, −dx_{1}|_{L}}. Then there is no vector valued function ξ such that ξ ∧ ℱ. On the other hand, |ℱ | = {ds|_{L}} and it is easy to see that M_{p,w}(|ℱ |) = M_{𝒜 ,p} (ℱ) = 0. For a special ℱ we can remove ‘(p, w)-a.e.’ and ‘(p′, w^{1−p′})-a.e.’ See (1.2) below.

The monotonicity of M_{𝒜 ,p} and M_{ℬ,p′} is clear from the definition. However, some of the known properties for M_{p,w} do not directly extend to M_{𝒜 ,p} and M_{ℬ,p′} . For example, the classical proof of the countable subadditivity does not apply to M_{𝒜 ,p} and M_{ℬ,p′} . The following proposition can be proved in a way common to the classical case and the present case. For completeness the proof will be given in Section 2.

Proposition 1. Let ℱ_{j} be an increasing sequence of vector measures such that ℱ = ⋃_{j} ℱ_{j}. Then

M_{𝒜 ,p} (ℱ_{j}) ↑ M_{𝒜 ,p} (ℱ).

Definition. Let K_{0} and K_{1} be disjoint compact subsets of R^{n} such that K_{0} ∩ \overline{Ω} ≠ ∅ and K_{1} ∩ \overline{Ω} ≠ ∅. We let

Γ = Γ(K_{0}, K_{1}, Ω)
= \{γ : curves in Ω starting from K_{0} ∩ \overline{Ω} and terminating at K_{1} ∩ \overline{Ω}\}.

We know that the family of all non-rectifiable curves is (p, w)-exc. Thus for (p, w)-a.e. γ ∈ Γ we have naturally a vector measure dx|_{γ} = (dx_{1}, …, dx_{n})|_{γ} and a measure ds|_{γ} = |dx| |_{γ}. We write d Γ = {dx|_{γ} : γ ∈ Γ} and |d Γ| = {ds|_{γ} : γ ∈ Γ}. More generally, for a positive definite symmetric matrix valued measurable function Q = \{q_{ij}\} we let |Qdx| |_{γ} = \sqrt{∑_{i,j} q_{ij} dx_{i} dx_{j}} |_{γ} and |Qd Γ| = \{|Qdx| |_{γ} : γ ∈ Γ}.

Hereafter, K_{0} and K_{1} will denote disjoint compact sets such that K_{0} ∩ \overline{Ω} ≠ ∅ and K_{1} ∩ \overline{Ω} ≠ ∅, unless otherwise specified. We can consider moduli M_{p,w}(|d Γ|), M_{𝒜 ,p} (d Γ) and M_{p}(|\sqrt{ℬ} d Γ|). The reciprocals of these quantities are called generalized extremal distances. By definition we may assume that K_{0}, K_{1} ⊂ \overline{Ω} by taking the intersection with \overline{Ω}.

We introduce a generalized capacity of condenser. A real function u on Ω is called a (p, w)-precise function if u is absolutely continuous on a (p, w)-a.e. curve in Ω and ∇u ∈ L_{w}^{p}(Ω). (From the first requirement ∇u exists a.e. in Ω. For details see Proposition 2 in Section 2.) Let 𝒟(K_{0}, K_{1}, Ω) be the family of all (p, w)-precise functions u on Ω such that u(x) tends to 0 as x → K_{0} ∩ \overline{Ω} along (p, w)-a.e. curve in Ω and that u(x) tends to 1 as x → K_{1} ∩ \overline{Ω} along (p, w)-a.e. curve in Ω. We define the capacity

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = inf_{u∈𝒟(K_{0},K_{1},Ω)} 𝒜_{p}(∇u)^{p}.

Let 𝒟^{∗} = 𝒟^{∗}(K_{0}, K_{1}, Ω) be the family of all (p, w)-precise functions u on Ω such that u = 0 on the intersection of Ω and a neighborhood of K_{0} and u = 1 on the intersection of Ω and a neighborhood of K_{1} . Similarly, we define the capacity

C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) = inf_{u∈𝒟^{∗}(K_{0},K_{1},Ω)} 𝒜_{p}(∇u)^{p}.

By definition 𝒟(K_{0}, K_{1}, Ω) ⊃ 𝒟^{∗}(K_{0}, K_{1}, Ω) ≠ ∅ and so

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) ≤ C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) < ∞.

We shall show that the capacity C_{𝒜 ,p} (K_{0}, K_{1}, Ω) agrees with the modulus M_{𝒜 ,p} .

Theorem 1.

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = M_{𝒜 ,p} (d Γ) = M_{p}(|\sqrt{ℬ} d Γ|) < ∞.

Remark. Let 𝒜′ be another matrix function such that 𝒜′ = 𝒜 a.e. and let ℬ′ be the inverse matrix of it. Then C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = C_{𝒜′,p} (K_{0}, K_{1}, Ω) and M_{𝒜 ,p} (d Γ) = M_{𝒜′,p} (d Γ) by definition. In general, the systems |\sqrt{ℬ} d Γ| and |\sqrt{ℬ′} d Γ| may differ. However, the above theorem says that their moduli always coincide. A similar remark will apply to Theorem 3 below.

We shall show that the capacity C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) is related to the modulus M_{ℬ,p′} . Let

∇𝒟^{∗} = ∇𝒟^{∗}(K_{0}, K_{1}, Ω) = {∇u : u ∈ 𝒟^{∗}(K_{0}, K_{1}, Ω)}

and consider M_{ℬ,p′} (∇𝒟^{∗}). We observe that ‘(p′, w^{1−p′})-a.e.’ in the definition of M_{ℬ,p′} can be removed for ℱ = ∇𝒟^{∗}. In view of Hölder’s inequality,

∫_{Ω} f|∇u| dx ≤ ∥∇u∥_{p,w} ∥f∥_{p′,w^{1−p′}} < ∞

for u ∈ 𝒟^{∗} and f ∈ L_{w^{1−p′}}^{p′} . Hence the property (vi) of M_{p,w} implies that only the empty set in ∇𝒟^{∗} is (p′, w^{1−p′})-exc. Therefore

(1.2)  M_{ℬ,p′} (∇𝒟^{∗}) = inf{ℬ_{p′}(ξ)^{p′} : ξ ∧ ∇𝒟^{∗}}.

With the aid of (1.2) and the minimax theorem of Ky Fan [3] (see also [1, Theorem 2.4.1]), we shall prove the following theorem.

Theorem 2. If C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) = 0, then we have M_{ℬ,p′} (∇𝒟^{∗}) = ∞; if C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) > 0, then

C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω)^{1/p}M_{ℬ,p′} (∇𝒟^{∗})^{1/p′} = 1.

In particular, 0 < M_{ℬ,p′} (∇𝒟^{∗}) ≤ ∞ in any case.

Let us consider families of sets separating K_{0} and K_{1} . More generally, we consider functions separating K_{0} and K_{1} . For this purpose we let BV (Ω) be the space of functions u on Ω of bounded variation. Here, a function u is said to be of bounded variation if u is locally integrable on Ω and its distributional derivative ∇u is a finite vector measure on Ω (see [8, Chapter 6], [5, Chapter 1], [12, Chapter 5] and [2, Chapter 5]). The total variation of ∇u is denoted by ∥u∥_{BV} and is calculated by

∥u∥_{BV} = sup{∫_{Ω} u div ξ dx : ξ ∈ C_{0}^{∞}(Ω; R^{n}), |ξ| ≤ 1}.

The perimeter of a set E relative to Ω is defined by P_{Ω}(E) = ∥χ_{E∩Ω}∥_{BV} if χ_{E∩Ω} ∈ BV (Ω) and P_{Ω}(E) = ∞ if χ_{E∩Ω} ∉ BV (Ω).

Definition. Let Ω be bounded. We let S = S(K_{0}, K_{1}, Ω) be the family of functions u ∈ BV (Ω) such that u = 0 on the intersection of Ω and a neighborhood of K_{0} and u = 1 on the intersection of Ω and a neighborhood of K_{1} . Let Σ = Σ(K_{0}, K_{1}, Ω) be the family of characteristic functions χ_{E} of sets E ⊂ Ω such that P_{Ω}(E) < ∞ and E = U ∩ Ω for some open set U ⊂ R^{n} with K_{0} ∩ \overline{U} = ∅ and K_{1} ⊂ U.

Remark. If Ω is unbounded, then unbounded sets “separating” K_{0} and K_{1} in Ω may be considered. In general, unbounded sets can have infinite perimeter. Thus the above definition of S(K_{0}, K_{1}, Ω) and Σ(K_{0}, K_{1}, Ω) has to be generalized. Such a generalization may be done by using the notions of “locally bounded variation” and “locally bounded perimeter” (see [2, Chapter 5]). For simplicity we shall restrict ourselves to a bounded domain Ω in this paper.

We define the following families:

∇S = ∇S(K_{0}, K_{1}, Ω) = {∇u : u ∈ S(K_{0}, K_{1}, Ω)},
|∇S| = |∇S(K_{0}, K_{1}, Ω)| = {|∇u| : u ∈ S(K_{0}, K_{1}, Ω)},
∇Σ = ∇Σ(K_{0}, K_{1}, Ω) = {∇χ_{E} : χ_{E} ∈ Σ(K_{0}, K_{1}, Ω)},
|∇Σ| = |∇Σ(K_{0}, K_{1}, Ω)| = {|∇χ_{E}| : χ_{E} ∈ Σ(K_{0}, K_{1}, Ω)}.

More generally, we let

\begin{aligned}|\sqrt{𝒜} ∇S| & = \{|\sqrt{𝒜} ∇u| : u ∈ S\}, \\ |\sqrt{𝒜} ∇Σ| & = \{|\sqrt{𝒜} ∇χ_{E}| : χ_{E} ∈ Σ\}. \end{aligned}

We consider M_{ℬ,p′} (∇S), M_{ℬ,p′} (∇Σ), M_{p′}(|\sqrt{𝒜} ∇S|), M_{p′}(|\sqrt{𝒜} ∇Σ|). The reciprocals of these quantities are called generalized extremal widths. Obviously, ∇Σ ⊂ ∇S, so that

M_{ℬ,p′} (∇S) ≥ M_{ℬ,p′} (∇Σ) and M_{p′}(|\sqrt{𝒜} ∇S|) ≥ M_{p′}(|\sqrt{𝒜} ∇Σ|).

There is a relationship between S and 𝒟^{∗} in Theorem 2. Since Ω is bounded and w ∈ A_{p}, it follows that ∫_{Ω} w^{1−p′} dx < ∞, so that by Hölder’s inequality

(1.3)  ∫_{Ω} |∇u| dx ≤ (∫_{Ω} |∇u|^{p}w dx)^{1/p}(∫_{Ω} w^{1−p′} dx)^{1/p′} < ∞

for any (p, w)-precise functions u. Hence 𝒟^{∗} ⊂ S and 0 < M_{ℬ,p′} (∇𝒟^{∗}) ≤ M_{ℬ,p′} (∇S). Moreover, we have the following theorem.

Theorem 3. Let Ω be bounded. Then

\begin{aligned}M_{ℬ,p′} (∇S) & = M_{ℬ,p′} (∇Σ) = M_{p′}(|\sqrt{𝒜} ∇S|) \\ & = M_{p′}(|\sqrt{𝒜} ∇Σ|) = M_{ℬ,p′} (∇𝒟^{∗}) > 0. \end{aligned}

We are interested in whether or not

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω).

If this is true, then Theorems 1–3 give a relationship between generalized extremal distances and extremal widths. We shall reduce this question to the problem of continuity of extremal distances. For this purpose we need the uniform continuity of w(x)^{1/p}\sqrt{ℬ(x)} , i.e., for any ε > 0, there exists δ > 0 such that if x, y ∈ Ω and |x − y| < δ, then

|w(x)^{1/p}\sqrt{ℬ(x)} − w(y)^{1/p}\sqrt{ℬ(y)} | < ε.

We remark that this continuity implies neither the continuity of \sqrt{ℬ(x)} nor w(x).

Theorem 4. Suppose that w(x)^{1/p}\sqrt{ℬ(x)} is uniformly continuous on Ω. Then M_{p}(|\sqrt{ℬ} d Γ|) has the continuity property. That is, let K_{0}^{j} and K_{1}^{j} be sequences of compact sets such that K_{0}^{0}∩K_{1}^{0} = ∅, K_{0}^{j} ⊂ int K_{0}^{j−1} , K_{1}^{j} ⊂ int K_{1}^{j−1} , K_{0} = ⋂_{j=0}^{∞} K_{0}^{j}, and K_{1} = ⋂_{j=0}^{∞} K_{1}^{j}. If Γ_{j} = Γ(K_{0}^{j}, K_{1}^{j}, Ω), then

M_{p}(|\sqrt{ℬ} d Γ_{j}|) ↓ M_{p}(|\sqrt{ℬ} d Γ|).

As a result of Theorems 1–4 we obtain the following reciprocal relation between extremal distance and extremal width.

Theorem 5. Suppose that w(x)^{1/p}\sqrt{ℬ(x)} is uniformly continuous on Ω. Then

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω).

Moreover, suppose Ω is bounded. If M_{𝒜 ,p} (d Γ) = 0, then M_{ℬ,p′} (∇Σ) = ∞; if M_{𝒜 ,p} (d Γ) > 0, then

M_{𝒜 ,p} (d Γ)^{1/p}M_{ℬ,p′} (∇Σ)^{1/p′} = 1.

The above theorem yields the classical reciprocal relation. Let 𝒜 = w^{2/p}E with the identity matrix E. Then \sqrt{𝒜} = w^{1/p}E, ℬ = w^{−2/p}E, \sqrt{ℬ} = w^{−1/p}E, M_{p}(|\sqrt{ℬ} d Γ|) = M_{p,w}(|d Γ|) and M_{p′}(|\sqrt{𝒜} ∇Σ|) = M_{p′,w^{1−p′}} (|∇Σ|). Hence we obtain

Corollary. Let Ω be bounded. If M_{p,w}(|d Γ|) = 0, then M_{p′,w^{1−p′}} (|∇Σ|) = ∞; if M_{p,w}(|d Γ|) > 0, then

M_{p,w}(|d Γ|)^{1/p}M_{p′,w^{1−p′}} (|∇Σ|)^{1/p′} = 1.

Acknowledgment. The authors would like to thank the referee for valuable comments.

2. Preliminaries

Proof of Proposition 1. Since M_{𝒜 ,p} is monotone, we may assume that

lim M_{𝒜 ,p} (ℱ_{j}) = α < ∞,

and we have only to show that M_{𝒜 ,p} (ℱ) ≤ α. By definition there is ξ_{j} such that ξ_{j} ∧ ℱ_{j} (p, w)-a.e. such that

M_{𝒜 ,p} (ℱ_{j}) ≤ 𝒜_{p}(ξ_{j})^{p} ≤ M_{𝒜 ,p} (ℱ_{j}) + \frac{1}{j}.

Let j ≤ i. We have \frac{1}{2}(ξ_{i}+ξ_{j})∧ℱ_{j} (p, w)-a.e. Hence 𝒜_{p}(\frac{1}{2}(ξ_{i}+ξ_{j})) ≥ M_{𝒜 ,p} (ℱ_{j}). Let us invoke the Clarkson inequalities. Suppose p ≥ 2. Then

𝒜_{p}(\frac{1}{2}(ξ_{i} − ξ_{j}))^{p} ≤ \frac{1}{2}𝒜_{p}(ξ_{i})^{p} + \frac{1}{2}𝒜_{p}(ξ_{j})^{p} − 𝒜_{p}(\frac{1}{2}(ξ_{i} + ξ_{j}))^{p}.

The first two terms on the right hand side tend to α and the limit of the last term is not less than α as i, j → ∞. Hence

∥\frac{1}{2}(ξ_{i} − ξ_{j})∥_{p,w} ≤ c_{0}𝒜_{p}(\frac{1}{2}(ξ_{i} − ξ_{j}))^{p} → 0.

Hence there is ξ such that ∥\frac{1}{2}(ξ − ξ_{j})∥_{p,w} → 0 and 𝒜_{p}(ξ)^{p} = α. The same is true of the case 1 < p ≤ 2. By the property (v) of M_{p,w}, taking a subsequence, if necessary, we obtain

∫ |ξ − ξ_{j}| d|ν| → 0 for (p, w)-a.e. ν ∈ ℱ .

Hence

∫ ξ·dν = ∫ ξ_{j}·dν+∫ (ξ−ξ_{j})·dν ≥ lim inf ∫ ξ_{j}·dν ≥ 1 for (p, w)-a.e. ν ∈ ℱ,

so that

M_{𝒜 ,p} (ℱ) ≤ 𝒜_{p}(ξ)^{p} = α.

The proposition is proved.

Let us clarify the meaning of ∇u for a (p, w)-precise function. To this end we introduce ACL (=absolutely continuous on lines) functions. We say that u is ACL if u is absolutely continuous on each component of the part in Ω of a.e. (= almost every with respect to (n − 1)-dimensional measure) line parallel to each coordinate axis. By definition a (p, w)-precise function u is ACL.

Proposition 2. Let u be an ACL function on Ω. Then the Dini derivatives ∂u/∂x_{i} exist a.e. in Ω and are measurable for i = 1, …, n. Moreover, if ∂u/∂x_{i} are locally integrable, then they coincide with the distributional derivatives of u.

Proof. In view of Nikodym [10, Lemme on p. 133], we see that u is measurable on Ω. For the first assertion it suffices to prove the existence and the measurability of ∂u/∂x_{i} on a closed cube Q ⊂ Ω. We may assume that i = 1. We find δ > 0 such that (x_{1} + h, x_{2}, …, x_{n}) ∈ Q whenever x = (x_{1}, …, x_{n}) ∈ Q and |h| < δ. Let

u_{h}(x) = \frac{u(x_{1} + h, x_{2}, …, x_{n}) − u(x_{1}, …, x_{n})}{h}

for |h| < δ. Since u( · , x_{2}, …, x_{n}) is absolutely continuous for a.e. (x_{2}, …, x_{n}), it follows that lim_{h→0} u_{h} exists a.e. on Q and is measurable.

Now suppose that ∂u/∂x_{i} is locally integrable on Ω. Take φ ∈ C_{0}^{∞}(Ω). Then φu is ACL on Ω and

0 = ∫_{−∞}^{+∞} \frac{∂(φu)}{∂x_{i}} dx_{i} = ∫_{−∞}^{+∞}(φ\frac{∂u}{∂x_{i}} + u\frac{∂φ}{∂x_{i}} ) dx_{i},

whence

∫_{Ω} φ\frac{∂u}{∂x_{i}} dx = − ∫_{Ω} u\frac{∂φ}{∂x_{i}} dx.

This means ∂u/∂x_{i} is the distributional derivative of u.

Observe that the last assertion of the proposition applies to a (p, w)-precise function u since w ∈ A_{p}, and so L_{w}^{p}(Ω) ⊂ L_{loc}^{1}(Ω). See [2, Section 4.9] for another proof of differentiability of a precise function. Note that a precise function is defined as a quasicontinuous function in [2].

In the rest of this section we shall deal with approximation. Let ψ be a nonnegative, radially symmetric smooth function supported in the unit ball {|y| < 1} such that ∫ ψ dy = 1. Let ψ_{r}(x) = r^{−n}ψ(x/r) and consider f ∗ψ_{r} for a locally integrable function f on R^{n}. It is well known that f ∗ψ_{r}(x) → f(x) as r → 0 and |f ∗ ψ_{r}(x)| ≤ cMf(x) for a.e. x ∈ R^{n}, where Mf is the maximal function of f and c is a positive constant depending only on ψ. Hence, if f ∈ L_{w}^{p}(R^{n}), then the maximal inequality ∥Mf∥_{p,w} ≤ c∥f∥_{p,w} and the dominated convergence theorem imply that ∥f ∗ ψ_{r} − f∥_{p,w} → 0 as r → 0. Thus f ∈ L_{w}^{p}(R^{n}) is approximated by smooth functions.

We would like to establish such an approximation property for functions defined only on an open set G ≠ R^{n}. To this end we use a positive smooth function α on G which vanishes on ∂G.

Lemma 2.1. There exists α = α_{G} ∈ C^{∞}(G) such that

(i) 0 < α ≤ 1, (ii) |∇α| ≤ \frac{1}{2} , (iii) α(x) ≤ \frac{1}{2} dist(x, ∂G).

Proof. Let \{G_{j}\}_{j=1}^{∞} be an increasing sequence of open sets such that \overline{G_{j}} ⋐ G_{j+1} and G_{j} ↑ G. For each j ≥ 1 we find a nonnegative function β_{j} ∈ C_{0}^{∞}(G) such that β_{j} > 0 on G_{j} \ G_{j−1} and β_{j} = 0 on (G \ G_{j+1}) ∪ G_{j−2} , where we put G_{0} = G_{−1} = ∅ conventionally. Multiplying by a positive constant, if necessary, we may assume that β_{j} ≤ \frac{1}{3} , |∇β_{j}| ≤ \frac{1}{6} and β_{j}(x) ≤ \frac{1}{6} dist(x, ∂G). Then the function α = ∑_{j=1}^{∞} β_{j} has the required properties.

Lemma 2.2. Let |y| ≤ 1 and 0 < r < 1. Then the mapping T_{y,r}: x → x + rα(x)y gives a C^{∞} homeomorphism of G onto itself.

Proof. If y = 0, then there is nothing to prove. Suppose y ≠ 0. In view of (iii) of Lemma 2.1 we see that T_{y,r} is a mapping from G to G. Suppose T_{y,r}(x) = T_{y,r}(x′) for x, x′ ∈ G. Then x − x′ = r(α(x) − α(x′))y, so that by (ii) of Lemma 2.1 |x − x′| ≤ \frac{1}{2}r|x − x′|, and hence x = x′. Thus T_{y,r} is injective. Let l be a component of the intersection of G and a line in the direction of y. Then, by the definition of T_{y,r} and (iii) of Lemma 2.1, l is invariant under T_{y,r}. In particular, T_{y,r} is a surjection onto G. Observe that the Jacobian of T_{y,r} is equal to 1 + r∇α · y ≥ 1 − \frac{1}{2}r > 0 by (ii) of Lemma 2.1. Therefore, the inverse mapping T_{y,r}^{−1} is C^{∞} too.

As before Lemma 2.1, let ψ be a nonnegative, radially symmetric smooth function supported in the unit ball {|y| < 1} such that ∫ ψ dy = 1. For a locally integrable function f on G we define

(2.1) (f)_{r}(x) = ∫_{|y|<1} f(x + rα(x)y)ψ(y) dy = \frac{1}{(rα(x))^{n}} ∫_{R^{n}} f(y)ψ(\frac{y − x}{rα(x)}) dy

for x ∈ G. Observe that (f)_{r} ∈ C^{∞}(G). It is easy to see that (f)_{r}(x) → f(x) as r → 0 and |(f)_{r}(x)| ≤ cMf(x) for a.e. x ∈ G. Hence the dominated convergence theorem yields

Proposition 3. Let f ∈ L_{w}^{p}(G). Then ∥(f)_{r} − f∥_{p,w} → 0 as r → 0.

With some calculation we can show the following proposition (cf. [9, Theorem 4.6]).

Proposition 4. Let u be a (p, w)-precise function over G. Then

∥∇(u)_{r} − ∇u∥_{p,w} → 0 as r → 0.

Let G = Ω and apply Proposition 4 to C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω). Suppose u ∈ 𝒟^{∗} = 𝒟^{∗}(K_{0}, K_{1}, Ω). By definition, if r > 0 is sufficiently small, then (u)_{r} ∈ 𝒟^{∗}. Hence, by Proposition 4 and truncation, we have

Proposition 5.

C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) = inf_{{u∈𝒟^{∗}∩C^{∞}(Ω) \atop 0≤u≤1}} 𝒜_{p}(∇u)^{p}.

3. Proof of Theorem 1

The proof of Theorem 1 will be based on the following three lemmas. Let us recall Γ = Γ(K_{0}, K_{1}, Ω). Without loss of generality we may assume that K_{0}, K_{1} ⊂ \overline{Ω}.

Lemma 3.1.

M_{𝒜 ,p} (d Γ) ≤ C_{𝒜 ,p} (K_{0}, K_{1}, Ω) < ∞.

Proof. Clearly 𝒟(K_{0}, K_{1}, Ω) ≠ ∅ so that C_{𝒜 ,p} (K_{0}, K_{1}, Ω) < ∞. Take u ∈ 𝒟(K_{0}, K_{1}, Ω). It is known (cf. [9, §§4.3–4]) that

∫_{γ} ∇u · dx = u(e(γ)) − u(s(γ)) = 1 for (p, w)-a.e. curve γ,

where s(γ) and e(γ) denote the starting point and the end point of γ, respectively. Hence ∇u ∧ d Γ (p, w)-a.e., so that 𝒜_{p}(∇u)^{p} ≥ M_{𝒜 ,p} (d Γ). Taking the infimum with respect to u, we obtain the lemma.

Lemma 3.2.

M_{p}(|\sqrt{ℬ} d Γ|) ≤ M_{𝒜 ,p} (d Γ) < ∞.

Proof. By Lemma 3.1 M_{𝒜 ,p} (d Γ) < ∞. Let ξ ∧ d Γ (p, w)-a.e. Then for (p, w)-a.e. γ ∈ Γ we have

1 ≤ ∫_{γ} ξ · dx = ∫_{γ} \sqrt{𝒜} ξ · \sqrt{ℬ} dx ≤ ∫_{γ} |\sqrt{𝒜} ξ||\sqrt{ℬ} dx|.

Thus |\sqrt{𝒜} ξ| ∧ |\sqrt{ℬ} d Γ| (p, w)-a.e. Taking the infimum with respect to ξ, we obtain

M_{𝒜 ,p} (d Γ) = inf 𝒜_{p}(ξ)^{p} = inf ∫_{Ω}|\sqrt{𝒜} ξ|^{p} dx ≥ M_{p}(|\sqrt{ℬ} d Γ|).

The lemma follows.

Lemma 3.3.

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) ≤ M_{p}(|\sqrt{ℬ} d Γ|) < ∞.

Proof. As was observed in the introduction,

M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ|) = M_{p}(|\sqrt{ℬ} d Γ|) < ∞.

Let ϱ be a nonnegative function in L_{w}^{p}(Ω) such that

(3.1)  ∫_{γ} ϱw^{1/p}|\sqrt{ℬ} dx| ≥ 1  for every curve γ ∈ Γ.

For each x ∈ Ω we let Γ_{0}^{x} be the family of curves starting in K_{0} and ending at x. Define

u(x) = inf_{γ∈Γ_{0}^{x}} ∫_{γ} ϱw^{1/p}|\sqrt{ℬ} dx|.

We claim

(i) u is (p, w)-precise in Ω; (ii) for a.e. x ∈ Ω

(3.2)  |\sqrt{𝒜 (x)} ∇u(x)| ≤ ϱ(x)w(x)^{1/p};

(iii) lim u(x) = 0 as x → K_{0} along (p, w)-a.e. curve γ ∈ Γ; (iv) lim inf u(x) ≥ 1 as x → K_{1} along (p, w)-a.e. curve γ ∈ Γ.

By (iv) of the properties of modulus stated in Section 1 and (1.1) we have

(3.3)  ∫_{γ} ϱw^{1/p}|\sqrt{ℬ} dx| ≤ c_{0} ∫_{γ} ϱ ds < ∞

for (p, w)-a.e. curve γ in Ω. Hence for (p, w)-a.e. curve γ

(3.4) |u(b) − u(a)| ≤ ∫_{ab} ϱw^{1/p}|\sqrt{ℬ} dx| ≤ c_{0} ∫_{ab} ϱ ds  for any points a, b ∈ γ,

where ãb is the arc on γ connecting a and b. This implies that u is absolutely continuous on γ for (p, w)-a.e. curve γ. We know that (p, w)-a.e. segment γ in Ω parallel to one of the coordinate axes satisfies (3.4) and hence by Fubini’s theorem |∂u/∂x_{i}| ≤ c_{0}ϱ a.e. in Ω for each i = 1, …, n. Hence ∇u ∈ L_{w}^{p}(Ω) and u is a (p, w)-precise function on Ω.

In order to show (3.2) we take a countable dense set {ξ_{j}} in the unit sphere {x : |x| = 1}. Then we see that (3.4) is satisfied along (p, w)-a.e. segment γ in Ω parallel to one of ξ_{j}. Hence

|u(a + hξ_{j}) − u(a)| ≤ ∫_{0}^{h} ϱ(a + tξ_{j})w(a + tξ_{j})^{1/p}|\sqrt{ℬ(a + tξ_{j})} ξ_{j}| dt

for a.e. a ∈ Ω. Divide by h and let h → 0. We have

(3.5)  |ξ_{j} · ∇u(a)| ≤ ϱ(a)w(a)^{1/p}|\sqrt{ℬ(a)} ξ_{j}|  for a.e. a ∈ Ω.

If |∇u(a)| = 0, then (3.2) is true at x = a. Suppose |∇u(a)| > 0. Then |\sqrt{𝒜 (a)} ∇u(a)| > 0. Take a sequence \{ξ_{j_{k}}\} tending to 𝒜 (a)∇u(a)/|𝒜 (a)∇u(a)|. Then by (3.5)

\begin{aligned}|\frac{𝒜 (a)∇u(a) · ∇u(a)}{|𝒜 (a)∇u(a)|} | & ≤ ϱ(a)w(a)^{1/p}|\frac{\sqrt{ℬ(a)} 𝒜 (a)∇u(a)}{|𝒜 (a)∇u(a)|} | \\ & = ϱ(a)w(a)^{1/p} \frac{|\sqrt{𝒜 (a)} ∇u(a)|}{|𝒜 (a)∇u(a)|}. \end{aligned}

Since |𝒜 (a)∇u(a) · ∇u(a)| = |\sqrt{𝒜 (a)} ∇u(a)|^{2} , we have (3.2) for x = a.

We have observed that a (p, w)-a.e. curve satisfies (3.3). For the proof of (iii) and (iv) take γ ∈ Γ satisfying (3.3) and express it in parameter: x_{γ}(t), t_{0} < t < t_{1} , such that x_{γ}(t) → K_{0} as t → t_{0} and x_{γ}(t) → K_{1} as t → t_{1} . By the definition of u(x) and (3.3)

0 ≤ u(x_{γ}(t)) ≤ ∫_{t_{0}}^{t} ϱw^{1/p}|\sqrt{ℬ} dx| ≤ c_{0} ∫_{t_{0}}^{t} ϱ ds → 0

as t → t_{0} . Thus lim u(x) = 0 as x → K_{0} along γ. We have (iii). Let us prove (iv) by contradiction. Suppose α = lim inf_{t→t_{1}} u(x_{γ}(t)) < 1 and let ε = 1 − α > 0. By definition there is t, t_{0} < t < t_{1} , such that

|u(x_{γ}(t)) − α| < \frac{ε}{3}  and  ∫_{t}^{t_{1}} ϱ ds < \frac{ε}{3c_{0}} .

By the definition of u(x) there is γ′ ∈ Γ_{0}^{x} with x = x_{γ}(t) such that

∫_{γ′} ϱw^{1/p}|\sqrt{ℬ} dx| < u(x) + \frac{ε}{3}.

Let γ′′ be the subcurve of γ corresponding to (t, t_{1}). Then γ′ + γ′′ ∈ Γ and by (3.3)

∫_{γ′+γ′′} ϱw^{1/p}|\sqrt{ℬ} dx| < u(x) + \frac{ε}{3} + c_{0} ∫_{t}^{t_{1}} ϱ ds < α + \frac{ε}{3} + \frac{ε}{3} + \frac{ε}{3} = 1,

which contradicts (3.1). Thus (iv) follows.

Now let ũ = min{u, 1}. Then ũ ∈ 𝒟(K_{0}, K_{1}, Ω). Hence by the definition of C_{𝒜 ,p} we have

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) ≤ 𝒜_{p}(∇ũ)^{p} ≤ 𝒜_{p}(∇u)^{p} = ∫_{Ω}|\sqrt{𝒜} ∇u|^{p} dx ≤ ∫_{Ω} ϱ^{p}w dx,

where (3.2) is used in the last inequality. Taking the infimum with respect to ϱ, we obtain

C_{𝒜 ,p} (K_{0}, K_{1}, Ω) ≤ M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ|) = M_{p}(|\sqrt{ℬ} d Γ|).

The lemma follows.

Combining Lemmas 3.1–3, we obtain Theorem 1.

4. Proof of Theorem 2

Proof of Theorem 2. We observe that 𝒟^{∗} is a convex set, Ξ = {ξ : ℬ_{p′}(ξ) ≤ 1} is a weakly compact convex set and

Φ(u, ξ) = − ∫_{Ω} ∇u · ξ dx

is a bilinear functional on 𝒟^{∗} × Ξ such that Φ(u, ·) is continuous with respect to the weak topology of Ξ. Hence the minimax theorem ([3]) yields

sup_{u∈𝒟^{∗}} inf_{ξ∈Ξ} Φ(u, ξ) = inf_{ξ∈Ξ} sup_{u∈𝒟^{∗}} Φ(u, ξ),

in other words

(4.1)  inf_{u∈𝒟^{∗}} sup_{ξ∈Ξ} ∫_{Ω} ∇u · ξ dx = sup_{ξ∈Ξ} inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx.

Observe that

(4.2)  sup_{ξ∈Ξ} ∫_{Ω} ∇u · ξ dx = 𝒜_{p}(∇u).

In fact, Hölder’s inequality yields

∫_{Ω} ∇u · ξ dx = ∫_{Ω} \sqrt{𝒜} ∇u · \sqrt{ℬ} ξ dx ≤ 𝒜_{p}(∇u)ℬ_{p′}(ξ) ≤ 𝒜_{p}(∇u)

for ξ ∈ Ξ. Conversely, let ξ = 𝒜_{p}(∇u)^{1−p}|\sqrt{𝒜} ∇u|^{p−2}A∇u. Then ℬ_{p′}(ξ) = 1 and ∫_{Ω} ∇u · ξ dx = 𝒜_{p}(∇u). Thus (4.2) follows, and (4.1) becomes

(4.3)  C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω)^{1/p} = sup_{ξ∈Ξ} inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx.

Suppose C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) = 0. Then inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx = 0 for any ξ ∈ Ξ, so that there is no ξ with ℬ_{p′}(ξ) < ∞ and ξ ∧ 𝒟^{∗}. Thus M_{ℬ,p′} (∇𝒟^{∗}) = ∞, and the theorem follows in this case. Now suppose C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) > 0. We claim

(4.4)  sup_{ξ∈Ξ} inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx = sup{ℬ_{p′}(ξ)^{−1} : inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx ≥ 1}.

Let α and β be the left hand side and the right hand side of (4.4), respectively. By definition

ℬ_{p′}(ξ) ≤ 1  ⟹  inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx ≤ α.

By homogeneity this is equivalent to

inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx > 1  ⟹  ℬ_{p′}(ξ) > \frac{1}{α}.

This means that β ≤ α. Similarly we can prove α ≤ β and we have (4.4). By definition the right hand side of (4.4) is equal to

(inf{ℬ_{p′}(ξ) : inf_{u∈𝒟^{∗}} ∫_{Ω} ∇u · ξ dx ≥ 1})^{−1} = M_{ℬ,p′} (∇𝒟^{∗})^{−1/p′},

where (1.2) is used. Hence (4.3) and (4.4) yield

C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω)^{1/p} = M_{ℬ,p′} (∇𝒟^{∗})^{−1/p′},

which yields the theorem.

5. Proof of Theorem 3

In this section we let Ω be bounded. The proof of Theorem 3 will be done by the following three lemmas.

Lemma 5.1.

(5.1)  M_{ℬ,p′} (∇S) ≥ M_{p′}(|\sqrt{𝒜} ∇S|) ≥ M_{p′}(|\sqrt{𝒜} ∇Σ|),
(5.2)  M_{ℬ,p′} (∇S) ≥ M_{ℬ,p′} (∇Σ) ≥ M_{p′}(|\sqrt{𝒜} ∇Σ|).

Proof. Let us prove the first inequality of (5.1). We may assume that M_{ℬ,p′} (∇S) < ∞. Let ξ ∧ ∇u for (p′, w^{1−p′})-a.e. ∇u ∈ ∇S. Then there is a (p′, w^{1−p′})-exc. subsystem 𝒰_{0} of ∇S such that ξ ∧ ∇u holds for every ∇u ∈ ∇S \ 𝒰_{0} . We have M_{p′,w^{1−p′}} (|𝒰_{0}|) = 0. We claim that M_{p′}(\{|\sqrt{𝒜} ∇u| : ∇u ∈ 𝒰_{0}}) = 0. In fact, for a given ε > 0, there is ϱ ≥ 0 such that ϱ ∧ |𝒰_{0}| and ∫ ϱ^{p′}w^{1−p′} dx < ε. By (1.1)

1 ≤ ∫ ϱ|∇u| = ∫ ϱw^{−1/p}w^{1/p}|∇u| ≤ c_{0} ∫ ϱw^{−1/p}|\sqrt{𝒜} ∇u|

for ∇u ∈ 𝒰_{0} . Hence

M_{p′}(\{|\sqrt{𝒜} ∇u| : ∇u ∈ 𝒰_{0}\}) ≤ ∫ (c_{0}ϱw^{−1/p})^{p′} dx = c_{0}^{p′} ∫ ϱ^{p′}w^{1−p′} dx < c_{0}^{p′}ε.

The arbitrariness of ε gives M_{p′}(\{|\sqrt{𝒜} ∇u| : ∇u ∈ 𝒰_{0}\}) = 0. For ∇u ∈ ∇S \ 𝒰_{0} we have

1 ≤ ∫_{Ω} ξ · ∇u = ∫_{Ω} \sqrt{ℬ} ξ · \sqrt{𝒜} ∇u ≤ ∫_{Ω}|\sqrt{ℬ} ξ||\sqrt{𝒜} ∇u|

so that |\sqrt{ℬ} ξ| ∧ |\sqrt{𝒜} ∇S| p′ -a.e. Taking the infimum with respect to ξ, we obtain

M_{ℬ,p′} (∇S) = inf ℬ_{p′}(ξ)^{p′} = inf ∫_{Ω}|\sqrt{ℬ} ξ|^{p′} dx ≥ M_{p′}(|\sqrt{𝒜} ∇S|).

Thus the first inequality of (5.1) follows. The second inequality of (5.1) and the first inequality of (5.2) are trivial. In the same way as above, we have the second inequality of (5.2). The lemma follows.

Lemma 5.2.

M_{p′}(|\sqrt{𝒜} ∇Σ|) ≥ M_{ℬ,p′} (∇𝒟^{∗}) > 0.

Proof. We may assume that M_{p′}(|\sqrt{𝒜} ∇Σ|) < ∞. In view of Theorem 2, it is sufficient to show that C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) > 0 and

(5.3)  C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω)^{1/p}M_{p′}(|\sqrt{𝒜} ∇Σ|)^{1/p′} ≥ 1.

Let u ∈ 𝒟^{∗} ∩ C^{∞}(Ω) and 0 ≤ u ≤ 1. Put N_{t} = {x ∈ Ω : u(x) > t}. We claim that χ_{N_{t}} ∈ Σ for a.e. t, 0 < t < 1. By the Sard theorem {x ∈ Ω : u(x) = t} is a smooth manifold for a.e. t. For such t Gauss’ divergence theorem yields

∫_{N_{t}} div φ dx = − ∫_{u=t} φ · n dS for φ ∈ C_{0}^{∞}(Ω; R^{n}),

where n = ∇u/|∇u| is the normal to {u = t} directed into N_{t}. By definition the above identity implies ∇χ_{N_{t}} = ndS|_{u=t}. By the coarea formula (see e.g. [8, Theorem 1.2.4])

∫_{0}^{1} ∥χ_{N_{t}}∥_{BV} dt = ∫_{0}^{1} dt ∫_{u=t} dS = ∫_{Ω} |∇u| dx.

The last integral is convergent by (1.3) and hence χ_{N_{t}} ∈ Σ for a.e. t.

Since M_{p′}(|\sqrt{𝒜} ∇Σ|) < ∞, there is ϱ ≥ 0 such that ϱ ∧ |\sqrt{𝒜} ∇Σ| and ∥ϱ∥_{p′} < ∞. Take such a ϱ. Then

∫_{u=t} ϱ|\sqrt{𝒜} n| dS ≥ 1 for a.e. t.

Hence the coarea formula yields

\begin{aligned}1 & ≤ ∫_{0}^{1} dt ∫_{u=t} ϱ|\sqrt{𝒜} n| dS = ∫_{0}^{1} dt ∫_{u=t} ϱ\frac{|\sqrt{𝒜} ∇u|}{|∇u|} dS \\ & = ∫_{Ω} ϱ|\sqrt{𝒜} ∇u| dx ≤ 𝒜_{p}(∇u) ∥ϱ∥_{p′} . \end{aligned}

Taking the infimum with respect to ϱ and u, we obtain C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) > 0 and (5.3) by Proposition 5. The lemma is proved.

The proof of Theorem 3 is completed by the following lemma.

Lemma 5.3.

M_{ℬ,p′} (∇𝒟^{∗}) ≥ M_{ℬ,p′} (∇S).

This is the most difficult part and the proof will be divided into several steps. In these steps, we shall vary the compact sets K_{0} , K_{1} and the open set Ω, so we shall write

\begin{aligned}M_{ℬ,p′} (K_{0}, K_{1}, Ω) & = M_{ℬ,p′} (∇S(K_{0}, K_{1}, Ω)), \\ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) & = M_{ℬ,p′} (∇𝒟^{∗}(K_{0}, K_{1}, Ω)). \end{aligned}

The outline of the proof is as follows: We prove, in Lemma 5.4, the monotonicity of M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) with respect to Ω and introduce an approximation property with respect to Ω. Lemma 5.5 shows the existence of an open subset of Ω with this approximation property. The approximation with respect to K_{0} and K_{1} stated in Lemma 5.6 is rather straightforward from Proposition 1. Lemma 5.7 includes the crucial part of the proof of Lemma 5.3, which deals with the case K_{0}, K_{1} ⊂ Ω under an additional assumption. The proof uses the approximation property of BV functions. Finally, by enlarging Ω, we reduce the general case K_{0}, K_{1} ⊂ \overline{Ω} to Lemma 5.7 and complete the proof of Lemma 5.3.

Lemma 5.4. Let Ω_{1} be an open subset of Ω such that K_{0}, K_{1} ⊂ \overline{Ω_{1}} . Then

M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) ≥ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω).

Proof. We may assume M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) < ∞. Let ξ ∧ ∇𝒟^{∗}(K_{0}, K_{1}, Ω_{1}) and set \overline{ξ} = ξ on Ω_{1} and \overline{ξ} = 0 on Ω \ Ω_{1} . Then

∫_{Ω} \overline{ξ} · ∇u dx = ∫_{Ω_{1}} ξ · ∇u|_{Ω_{1}} dx ≥ 1

for u ∈ 𝒟^{∗}(K_{0}, K_{1}, Ω) since u|_{Ω_{1}} ∈ 𝒟^{∗}(K_{0}, K_{1}, Ω_{1}). Hence

∫_{Ω_{1}} B[ξ]^{p′} dx = ℬ_{p′}(\overline{ξ})^{p′} ≥ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω).

Taking the infimum with respect to ξ, we obtain the lemma.

In view of the above monotonicity we introduce the following definition.

Definition. Let K_{0}, K_{1} ⊂ Ω. We say that Ω can be approximated from inside with respect to M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) if for each ε > 0 there is an open set Ω′ such that K_{0}, K_{1} ⊂ Ω′ ⊂ \overline{Ω′} ⊂ Ω and M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω′) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) + ε.

Lemma 5.5. Let K_{0}, K_{1} ⊂ Ω_{1} ⊂ \overline{Ω_{1}} ⊂ Ω. Suppose that M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) < ∞. Then there is Ω_{2} such that \overline{Ω_{1}} ⊂ Ω_{2} ⊂ Ω and Ω_{2} can be approximated from inside with respect to M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{2}).

Proof. For small r > 0 we let

Ω(r) = {x ∈ Ω : dist(x, ∂ Ω) > r}.

Observe that Ω(r) ↑ Ω as r ↓ 0. There is an r_{1} > 0 such that \overline{Ω_{1}} ⊂ Ω(r_{1}). Put φ(r) = M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω(r)). By Lemma 5.4 φ(r) is a nondecreasing function with 0 < φ(r) ≤ φ(r_{1}) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) < ∞ for 0 < r ≤ r_{1} , whence φ(r) is continuous from the right at r outside a countable set of r. Hence, there is r_{2} , 0 < r_{2} < r_{1} , such that lim_{r↓r_{2}} φ(r) = φ(r_{2}), i.e.,

M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω(r)) ↓ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω(r_{2})) as r ↓ r_{2}.

If Ω_{2} = Ω(r_{2}), we observe that \overline{Ω_{1}} ⊂ Ω_{2} ⊂ Ω and Ω_{2} can be approximated from inside with respect to M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{2}).

Lemma 5.6. Let K_{0}, K_{1} ⊂ Ω and let {K_{0}^{j}} and {K_{1}^{j}} be decreasing sequences of compact sets such that Ω ⊃ K_{0}^{j} ↓ K_{0} and Ω ⊃ K_{1}^{j} ↓ K_{1} . Then

M_{ℬ,p′} (K_{0}^{j}, K_{1}^{j}, Ω) ↑ M_{ℬ,p′} (K_{0}, K_{1}, Ω).

Proof. It is easy to see that

∇S(K_{0}^{j}, K_{1}^{j}, Ω) ↑ ∇S(K_{0}, K_{1}, Ω).

Hence the lemma follows from Proposition 1.

Lemma 5.7. Let K_{0}, K_{1} ⊂ Ω and suppose Ω can be approximated from inside with respect to M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω). Then

M_{ℬ,p′} (K_{0}, K_{1}, Ω) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω).

Proof. Let ε > 0. By assumption there exists Ω_{1} such that K_{0}, K_{1} ⊂ Ω_{1} ⊂ \overline{Ω_{1}} ⊂ Ω and

(5.4)  M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) + ε.

By Lemma 5.6 there are compact sets K_{0}^{1} and K_{1}^{1} such that K_{0} ⊂ int K_{0}^{1} ⊂ K_{0}^{1} ⊂ Ω, K_{1} ⊂ int K_{1}^{1} ⊂ K_{1}^{1} ⊂ Ω, and

(5.5)  M_{ℬ,p′} (K_{0}, K_{1}, Ω) ≤ M_{ℬ,p′} (K_{0}^{1}, K_{1}^{1}, Ω) + ε.

We claim

(5.6)  M_{ℬ,p′} (K_{0}^{1}, K_{1}^{1}, Ω) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}).

Then the required inequality follows from (5.4), (5.5), (5.6) and the arbitrariness of ε > 0.

Let us prove (5.6). We may assume that M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω_{1}) < ∞. Let ξ ∧ ∇𝒟^{∗}(K_{0}, K_{1}, Ω_{1}). We extend ξ by \overline{ξ} = ξ on Ω_{1} and \overline{ξ} = 0 elsewhere. Let ξ_{r} be the symmetric mollification, i.e.,

ξ_{r}(x) = ∫_{|y|<1} \overline{ξ}(x + ry)ψ(y) dy,

where ψ is a nonnegative, radially symmetric smooth function supported in the unit ball {|y| < 1} such that ∫ ψ dy = 1. Let

Ω_{2} = \{x ∈ Ω : dist(x, ∂ Ω) > \frac{1}{2} dist(\overline{Ω_{1}}, ∂ Ω)\}

and r_{0} = min\{dist(K_{0}, ∂K_{0}^{1}), dist(K_{1}, ∂K_{1}^{1}), \frac{1}{2} dist(\overline{Ω_{1}}, ∂ Ω)\} > 0. Let 0 < r < r_{0} . We shall see that

(5.7)  ξ_{r} ∈ C_{0}^{∞}(Ω_{2}; R^{n})  and  ξ_{r} ∧ ∇𝒟^{∗}(K_{0}^{1}, K_{1}^{1}, Ω).

The first assertion of (5.7) follows from the definition. Suppose u ∈ 𝒟^{∗}(K_{0}^{1}, K_{1}^{1}, Ω). Then

\begin{aligned}∫_{Ω} ξ_{r} · ∇u dx & = ∫_{|y|<1} ψ(y) dy ∫_{Ω} \overline{ξ}(x + ry) · ∇u(x) dx \\ & = ∫_{|y|<1} ψ(y) dy ∫_{Ω_{1}} ξ(x) · ∇u(x − ry) dx. \end{aligned}

Observe that u(x−ry)|_{Ω_{1}} ∈ 𝒟^{∗}(K_{0}, K_{1}, Ω_{1}) for |y| < 1. As ξ ∧∇𝒟^{∗}(K_{0}, K_{1}, Ω_{1}), it follows that ∫_{Ω_{1}} ξ(x) · ∇u(x − ry) dx ≥ 1, so that ∫_{Ω} ξ_{r} · ∇u dx ≥ 1. Hence (5.7) follows.

Next we claim that

(5.8)  ξ_{r} ∧ ∇S(K_{0}^{1}, K_{1}^{1}, Ω).

It is well known that a BV function is approximated by smooth functions (see e.g. [8, Theorem 6.1.2]). Let u ∈ S(K_{0}^{1}, K_{1}^{1}, Ω). Then we find a sequence of smooth functions u_{j} ∈ C^{∞}(Ω) such that u_{j} = 0 on a neighborhood of K_{0}^{1} , u_{j} = 1 on a neighborhood of K_{1}^{1} , u_{j} → u in L^{1}(Ω), ∥u_{j}∥_{BV} → ∥u∥_{BV} and ∇u_{j} → ∇u weakly, i.e.,

∫_{Ω} φ · ∇u_{j} dx → ∫_{Ω} φ · ∇u

for any φ ∈ C_{0}^{∞}(Ω; R^{n}). Note that the sign ‘dx’ does not appear on the right hand side, since ∇u is a measure which need not be absolutely continuous with respect to the Lebesgue measure. Observe that it may happen that ∫_{Ω} |∇u_{j}|^{p}w dx = ∞. To avoid this difficulty, take f ∈ C_{0}^{∞}(Ω) such that 0 ≤ f ≤ 1 on Ω and f = 1 on Ω_{2} . Then we see that fu_{j} ∈ C_{0}^{∞}(Ω) and hence fu_{j} ∈ 𝒟^{∗}(K_{0}^{1}, K_{1}^{1}, Ω). Hence (5.7) yields

1 ≤ ∫_{Ω} ξ_{r} · ∇(fu_{j}) dx = ∫_{Ω} ξ_{r} · ∇u_{j} dx → ∫_{Ω} ξ_{r} · ∇u.

Thus (5.8) follows.

By (5.8) we have M_{ℬ,p′} (K_{0}^{1}, K_{1}^{1}, Ω) ≤ ℬ_{p′}(ξ_{r})^{p′}, and by the approximation property, the right hand side tends to ℬ_{p′}(ξ)^{p′} as r ↓ 0. Hence, taking the infimum with respect to ξ we obtain (5.6). Thus the lemma is proved.

Proof of Lemma 5.3. We may assume that

M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω) = M_{ℬ,p′} (∇𝒟^{∗}) < ∞.

If necessary, taking the intersection with \overline{Ω}, we may assume that K_{0}, K_{1} ⊂ \overline{Ω}. Let us take decreasing sequences {K_{0}^{j}} and {K_{1}^{j}} of compact sets such that K_{0}^{j+1} ⊂ int K_{0}^{j}, K_{1}^{j+1} ⊂ int K_{1}^{j}, K_{0}^{j} ↓ K_{0} and K_{1}^{j} ↓ K_{1} . Let Ω^{j} = Ω ∪ (int K_{0}^{j}) ∪ (int K_{1}^{j}). We claim

(5.9)  ∇S(K_{0}^{j}, K_{1}^{j}, Ω^{j}) ↑ ∇S(K_{0}, K_{1}, Ω).

Take u ∈ S(K_{0}^{j}, K_{1}^{j}, Ω^{j}). By ∇_{Ω^{j}}u we denote the distributional derivative of u over Ω^{j}. By definition ∇_{Ω^{j}}u is concentrated on Ω^{j} \ (K_{0}^{j} ∪ K_{1}^{j}) ⊂ Ω and ∇_{Ω^{j}}u = ∇_{Ω^{j+1}}(u|_{Ω^{j+1}}) = ∇_{Ω}(u|_{Ω}). Hence ∇S(K_{0}^{j}, K_{1}^{j}, Ω^{j}) is increasing and ⋃ ∇S(K_{0}^{j}, K_{1}^{j}, Ω^{j}) ⊂ ∇S(K_{0}, K_{1}, Ω). Conversely, take u ∈ S(K_{0}, K_{1}, Ω). By definition if j is sufficiently large, then u = 0 on the intersection of Ω and a neighborhood of K_{0}^{j} and u = 1 on the intersection of Ω and a neighborhood of K_{0}^{j}. Let

\overline{u} = \{ \begin{matrix}u & on Ω, \\ 0 & on int K_{0}^{j}, \\ 1 & on int K_{1}^{j}. \end{matrix}

Then \overline{u} ∈ S(K_{0}^{j}, K_{1}^{j}, Ω^{j}), ∇_{Ω^{j}} \overline{u} is concentrated on Ω \ (K_{0}^{j} ∪ K_{1}^{j}), and ∇_{Ω^{j}} \overline{u} = ∇_{Ω}u. Hence ∇_{Ω}u ∈ ∇S(K_{0}^{j}, K_{1}^{j}, Ω^{j}). Thus (5.9) follows.

With the aid of Proposition 1 and (5.9) we have

(5.10)  M_{ℬ,p′} (K_{0}^{j}, K_{1}^{j}, Ω^{j}) ↑ M_{ℬ,p′} (K_{0}, K_{1}, Ω).

Since Ω ⊂ Ω^{j}, it follows from Lemma 5.4 that

M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω^{j}) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω),

where the right hand side has been assumed to be finite. Hence, in view of Lemma 5.5, we can modify the above K_{0}^{j}, K_{1}^{j} and Ω^{j} so that Ω^{j} can be approximated from inside with respect to M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω^{j}). Obviously S(K_{0}^{j}, K_{1}^{j}, Ω^{j}) ⊂ S(K_{0}, K_{1}, Ω^{j}) and so M_{ℬ,p′} (K_{0}^{j}, K_{1}^{j}, Ω^{j}) ≤ M_{ℬ,p′} (K_{0}, K_{1}, Ω^{j}). Now by (5.10) and Lemmas 5.4 and 5.7 we have

M_{ℬ,p′} (K_{0}, K_{1}, Ω) = lim M_{ℬ,p′} (K_{0}^{j}, K_{1}^{j}, Ω^{j}) ≤ lim M_{ℬ,p′} (K_{0}, K_{1}, Ω^{j})
≤ lim M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω^{j}) ≤ M_{ℬ,p′}^{∗} (K_{0}, K_{1}, Ω).

Thus the lemma follows.

Combining all the Lemmas 5.1–5.3, we obtain Theorem 3.

6. Proof of Theorems 4 and 5

As was observed in the introduction, we have

M_{p}(|\sqrt{ℬ} d Γ|) = M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ|).

Moreover, by (1.1)

c_{0}^{−1} ds ≤ w(x)^{1/p}|\sqrt{ℬ} dx| ≤ c_{0} ds

for a rectifiable curve γ, dx = dx|_{γ} and ds = |dx|. Throughout this section we let dσ = w(x)^{1/p}|\sqrt{ℬ} dx| for each rectifiable curve. We have c_{0}^{−1} ≤ dσ/ds ≤ c_{0} . The following lemma depends on the ingenious idea of Shlyk ([11]) which was digested by Ohtsuka ([9, Chapter 2]).

Lemma 6.1 (Shlyk–Ohtsuka). Let ϱ ∈ L_{w}^{p}(R^{n}) be a positive lower semicontinuous function which is continuous on Ω \ (K_{0} ∪ K_{1}). Let K_{0}^{j} and K_{1}^{j} be sequences of compact sets such that K_{0}^{j} ⊂ int K_{0}^{j−1} , K_{1}^{j} ⊂ int K_{1}^{j−1} , K_{0} = ⋂_{j=0}^{∞} K_{0}^{j}, and K_{1} = ⋂_{j=0}^{∞} K_{1}^{j}. Then for each ε > 0 we can construct a function ϱ′ on Ω, ϱ′ ≥ ϱ, with the following properties

(i) ∫_{Ω} ϱ′^{p}w dx ≤ ∫_{Ω} ϱ^{p}w dx + ε. (ii) Suppose for each j there is γ_{j} ∈ Γ_{j} = Γ(K_{0}^{j}, K_{1}^{j}, Ω) such that ∫_{γ_{j}} ϱ′ dσ ≤ α.

Then there exists γ̃ ∈ Γ(K_{0}, K_{1}, Ω) such that ∫_{γ} ϱ dσ ≤ α + ε.

Remark. The most difficult part of the lemma is the existence of γ̃ inside Ω. It is rather easy to find a curve in the closure of Ω with the integral inequality of (ii). However, such a curve need not belong to Γ(K_{0}, K_{1}, Ω).

Proof of Lemma 6.1. The construction of ϱ′ is as follows. Let K^{j} = K_{0}^{j} ∪K_{1}^{j}, W^{j} = K^{j−1}\int K^{j} and d_{j} = dist(∂K^{j−1}, ∂K^{j}) > 0. By assumption inf_{W^{j}∩Ω} ϱ > 0 and so we can find ε_{j} ↓ 0 such that

(6.1)  ∑_{j=1}^{∞}(1 + ε_{j}^{−1})^{p}ε_{j}^{p+1} < ε,
(6.2)  c_{0}αε_{j} < d_{j} inf_{W^{j}∩Ω} ϱ.

We can find a sequence of compact subsets Ω_{j} of Ω increasing to Ω such that

∫_{Ω\Ω_{j}} ϱ^{p}w dx < ε_{j}^{p+1}.

Let V^{j} = (Ω \ Ω_{j}) ∩ W^{j} and set

ϱ′(x) = \{ \begin{matrix}(1 + ε_{j}^{−1})ϱ(x) & if x ∈ V^{j}, \\ ϱ(x) & if x ∈ Ω \ (∪V^{j}). \end{matrix}

It is easy to see that (i) holds. In fact, observe from (6.1) that

\begin{aligned}∫_{Ω} ϱ′^{p}w dx & = ∑_{j} ∫_{V^{j}}[(1 + ε_{j}^{−1})ϱ]^{p}w dx + ∫_{Ω\(∪V^{j})} ϱ^{p}w dx \\ & ≤ ∑_{j}(1 + ε_{j}^{−1})^{p}ε_{j}^{p+1} + ∫_{Ω} ϱ^{p}w dx < ε + ∫_{Ω} ϱ^{p}w dx. \end{aligned}

Now let us show (ii). For a moment we fix j ≥ 1. By definition γ_{k} ∈ Γ_{j} = Γ(K_{0}^{j}, K_{1}^{j}, Ω) for k ≥ j. Hence γ_{k} includes arcs γ′_{k} and γ′′_{k} such that γ′_{k} connects ∂K_{0}^{j} and ∂K_{0}^{j−1} ; γ′′_{k} connects ∂K_{1}^{j} and ∂K_{1}^{j−1} . We claim that γ′_{k} and γ′′_{k} are not included in V^{j}. In fact, if γ′_{k} ⊂ V^{j}, then we would have from (6.2)

α ≥ ∫_{γ_{k}} ϱ′ dσ ≥ ∫_{γ′_{k}} ϱ′ dσ ≥ ε_{j}^{−1} ∫_{γ′_{k}} ϱ dσ ≥ ε_{j}^{−1}c_{0}^{−1}d_{j} inf_{W^{j}∩Ω} ϱ > α,

a contradiction. Therefore

γ_{k} ∩ (Ω_{j} ∩ (K_{0}^{j−1} \ int K_{0}^{j})) ≠ ∅  and  γ_{k} ∩ (Ω_{j} ∩ (K_{1}^{j−1} \ int K_{1}^{j})) ≠ ∅

for k ≥ j. Observe that Ω_{j} ∩ (K_{0}^{j−1} \ int K_{0}^{j}) is compact. We can find a point x_{0}^{j} ∈ Ω_{j} ∩(K_{0}^{j−1} \int K_{0}^{j}) such that for some subsequence {γ_{k}^{j}}_{k} there is a point in γ_{k}^{j} converging to x_{0}^{j}. Hence, if we place a small closed ball B_{0}^{j} ⊂ Ω with center at x_{0}^{j}, then we may assume that the subsequence γ_{k}^{j} eventually intersects B_{0}^{j}. Since ϱ is continuous at x_{0}^{j} ∈ Ω \ K^{j}, we can choose B_{0}^{j} so small that

(6.3)  ∫_{l} ϱ dσ ≤ ε/2^{j+3}  for any segment l ⊂ B_{0}^{j}.

In the same way we can choose a small closed ball B_{1}^{j} ⊂ Ω with center at x_{1}^{j} ∈ Ω_{j} ∩ (K_{1}^{j−1} \ int K_{1}^{j}) so that γ_{k}^{j} intersects B_{1}^{j}. We start this process from j = 1 and choose subsequences {γ_{k}^{j}}_{k} inductively such that γ_{k}^{j} intersects B_{0}^{j} and B_{1}^{j}. See Figure 1.

Figure 1.

Now let us consider the diagonal γ_{k}^{k}. Then γ_{k}^{k} intersects B_{0}^{j} and B_{1}^{j} for 1 ≤ j ≤ k. To γ_{k}^{k} we add two suitable radii of B_{i}^{j} for each i = 0, 1 and for 1 ≤ j ≤ k so that we have a connected curve γ̃_{k} ∈ Γ(K_{0}^{k}, K_{1}^{k}, Ω) crossing over {x_{0}^{j}, x_{1}^{j}}_{j=1}^{k}. By (6.3) we have

∫_{γ_{k}} ϱ dσ ≤ ∫_{γ_{k}^{k}} ϱ dσ + 2 ∑_{j=1}^{k} \frac{ε}{2^{j+3}} ≤ α + \frac{ε}{4}.

Let Γ_{0} be the totality of all curves in Ω \ (K_{0} ∪ K_{1}) connecting x_{0}^{1} and x_{1}^{1} . For i = 0, 1 let Γ_{i}^{j} be the totality of all curves in Ω \ (K_{0} ∪ K_{1}) connecting x_{i}^{j} and x_{i}^{j+1} . Then

inf_{γ∈Γ_{0}} ∫_{γ} ϱ dσ + ∑_{j=1}^{k} inf_{γ∈Γ_{0}^{j}} ∫_{γ} ϱ dσ + ∑_{j=1}^{k} inf_{γ∈Γ_{1}^{j}} ∫_{γ} ϱ dσ ≤ ∫_{γ_{k}} ϱ dσ ≤ α + \frac{ε}{4}.

Therefore we can choose C_{0} ∈ Γ_{0} and C_{i}^{j} ∈ Γ_{i}^{j} such that

∫_{C_{0}} ϱ dσ < inf_{γ∈Γ_{0}} ∫_{γ} ϱ dσ + \frac{ε}{2},
∫_{C_{i}^{j}} ϱ dσ < inf_{γ∈Γ_{i}^{j}} ∫_{γ} ϱ dσ + \frac{ε}{2^{j+3}} .

Let

γ̃ = ⋯ + C_{0}^{1} + C_{0} + C_{1}^{1} + · · · .

See Figure 2.

Figure 2.

Then γ̃ ∈ Γ(K_{0}, K_{1}, Ω) and

∫_{γ} ϱ dσ ≤ α + \frac{ε}{4} + \frac{ε}{2} + 2 ∑_{j=1}^{∞} \frac{ε}{2^{j+3}} = α + ε.

The lemma is proved.

Proof of Theorem 4. Since M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ|) = M_{p}(|\sqrt{ℬ} d Γ|), it is sufficient to show that

(6.4)  M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ_{j}|) ↓ M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ|).

Let M be the right hand side and let 0 < ε < \frac{1}{2} . By definition there is a nonnegative function ϱ̃ such that ϱ̃ ∧ w^{1/p}|\sqrt{ℬ} d Γ| and \left∥{ \atop ϱ̃}\right∥_{p,w}^{p} < M + ε. If necessary, adding a positive function on R^{n}, we may assume that ϱ̃ is strictly positive on R^{n}. Moreover, by the Vitali–Carathéodry theorem, we may assume that ϱ̃ is lower semicontinuous on R^{n}. Let

(ϱ̃)_{r}(x) = ∫_{|y|<1} ϱ̃(x + rα(x)y)ψ(y) dy

be as in (2.1) with G = Ω \ (K_{0} ∪ K_{1}). We observe that (ϱ̃)_{r} is C^{∞}(G) and in particular continuous on G. If r is sufficiently small, then ∥(ϱ̃)_{r}∥_{p,w}^{p} < M + ε.

By the uniform continuity of w(x)^{1/p}\sqrt{ℬ(x)} , we can take δ > 0 such that if x, z ∈ Ω and |x − z| < δ, then

(6.5)  |w(x)^{1/p}\sqrt{ℬ(x)} − w(z)^{1/p}\sqrt{ℬ(z)} | < c_{0}^{−1}ε,

where c_{0} is the constant in (1.1). Let 0 < r < δ. We claim

(6.6)  (1 + ε)(ϱ̃)_{r} ∧ w^{1/p}|\sqrt{ℬ} d Γ|.

Let γ ∈ Γ. We have

(6.7)  \begin{aligned}∫_{γ}(ϱ̃)_{r} dσ & = ∫_{|y|<1} ψ(y) dy ∫_{γ} ϱ̃(x + rα(x)y) dσ(x) \\ & = ∫_{|y|<1} ψ(y) dy ∫_{T_{y,r}(γ)} ϱ̃(z) dσ(T_{y,r}^{−1}(z)), \end{aligned}

where T_{y,r}(x) = x + rα(x)y. For a moment we fix y, |y| < 1. By the property of α(x) we see that T_{y,r}(γ) ∈ Γ and so ϱ̃ ∧ dσ|_{T_{y,r}(γ)} . Since |x − z| < r < δ with z = T_{y,r}(x), it follows from (1.1) and (6.5) that

\begin{aligned}|w(z)^{1/p}\sqrt{ℬ(z)} ξ| & ≤ |w(x)^{1/p}\sqrt{ℬ(x)} ξ| + |(w(x)^{1/p}\sqrt{ℬ(x)} − w(z)^{1/p}\sqrt{ℬ(z)} )ξ| \\ & ≤ |w(x)^{1/p}\sqrt{ℬ(x)} ξ| + c_{0}^{−1}ε|ξ| ≤ (1 + ε)|w(x)^{1/p}\sqrt{ℬ(x)} ξ| \end{aligned}

for any vectors ξ ∈ R^{n}. Hence

(1 + ε) dσ(T_{y,r}^{−1}(z)) ≥ dσ(z) for z ∈ T_{y,r}(γ),

so that by (6.7) and ϱ̃ ∧ dσ|_{T_{y,r}(γ)} ,

(1 + ε) ∫_{γ}(ϱ̃)_{r} dσ ≥ ∫_{|y|<1} ψ(y) dy ∫_{T_{y,r}(γ)} ϱ̃(z) dσ(z) ≥ 1.

Thus (6.6) is proved.

Now let ϱ = (1 + ε)(ϱ̃)_{r} with sufficiently small r > 0. Then

(6.8)  ∫_{γ} ϱ dσ ≥ 1  for any γ ∈ Γ,
(6.9)  ∫_{Ω} ϱ^{p}w dx < (1 + ε)^{p}(M + ε).

Let ϱ′ be as in Lemma 6.1. We show that

∫_{γ} ϱ′ dσ > 1 − 2ε for γ ∈ Γ_{j}

with sufficiently large j. In fact, suppose the contrary. Then there would be a sequence {j_{k}} and curves γ_{k} ∈ Γ_{j_{k}} such that

∫_{γ_{k}} ϱ′ dσ ≤ 1 − 2ε.

By Lemma 6.1 we would find γ̃ ∈ Γ such that

∫_{γ} ϱ dσ ≤ 1 − 2ε + ε = 1 − ε,

a contradiction to (6.8).

Now the proof is easy. We have (1 − 2ε)^{−1}ϱ′ ∧ w^{1/p}|\sqrt{ℬ} d Γ_{j}| for sufficiently large j and so

M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ_{j}|) ≤ ∫_{Ω}[(1 − 2ε)^{−1}ϱ′]^{p}w dx ≤ (1 − 2ε)^{−p}((1 + ε)^{p}(M + ε) + ε)

by (6.9). Hence, letting j → ∞ and then ε → 0, we obtain

lim sup_{j→∞} M_{p,w}(w^{1/p}|\sqrt{ℬ} d Γ_{j}|) ≤ M,

which yields (6.4). The theorem is proved.

For the proof of Theorem 5 we prepare two lemmas. With a slight abuse of notation, we say that a set X ⊂ Ω is (p, w)-exc. if M_{p,w}(Γ_{X}) = 0, where Γ_{X} is the family of curves in Ω which terminate at X. We say that a property holds (p, w)-a.e. if the set of points where the property fails to hold is (p, w)-exc. We shall employ the following lemmas given by Ohtsuka [9, Chapter 4].

Lemma 6.2. Let u be a (p, w)-precise function in Ω and let Γ be a family of curves in Ω whose end points lie in Ω. Then u tends to the value of u at the end point along (p, w)-a.e. curve of Γ.

Lemma 6.3. Let u be a (p, w)-precise function in Ω. Then any function equal to u (p, w)-a.e. in Ω is (p, w)-precise in Ω.

Proof of Theorem 5. Let K_{0}^{j} and K_{1}^{j} be as in Theorem 4. Take u ∈ 𝒟(K_{0}^{j}, K_{1}^{j}, Ω). We put

\overline{u} = \{ \begin{matrix}0 & on K_{0}^{j} ∩ Ω, \\ 1 & on K_{1}^{j} ∩ Ω, \\ u & on Ω \ (K_{0}^{j} ∪ K_{1}^{j}). \end{matrix}

By the definition of 𝒟(K_{0}^{j}, K_{1}^{j}, Ω) and Lemma 6.2 we have u = 0 (p, w)-a.e. on K_{0}^{j}∩Ω and u = 1 (p, w)-a.e. on K_{1}^{j}∩Ω. Hence u = \overline{u} (p, w)-a.e. in Ω and hence \overline{u} is a (p, w)-precise function in Ω by Lemma 6.3. By definition \overline{u} ∈ 𝒟^{∗}(K_{0}, K_{1}, Ω). Hence C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) ≤ 𝒜_{p}(∇\overline{u}) = 𝒜_{p}(∇u), and taking the infimum with respect to u, we obtain C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) ≤ C_{𝒜 ,p} (K_{0}^{j}, K_{1}^{j}, Ω). By Theorem 1 we have C_{𝒜 ,p} (K_{0}^{j}, K_{1}^{j}, Ω) = M_{p}(|\sqrt{ℬ} d Γ(K_{0}^{j}, K_{1}^{j}, Ω)|) and C_{𝒜 ,p} (K_{0}, K_{1}, Ω) = M_{p}(|\sqrt{ℬ} d Γ|). By Theorem 4 the first quantity tends to the second as j → ∞. Hence C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) ≤ C_{𝒜 ,p} (K_{0}, K_{1}, Ω). The opposite inequality is obvious and we have C_{𝒜 ,p}^{∗} (K_{0}, K_{1}, Ω) = C_{𝒜 ,p} (K_{0}, K_{1}, Ω). The second assertion of the theorem readily follows from Theorems 1–3.

88 Hiroaki Aikawa and Makoto Ohtsuka

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Received 14 April 1997