Characterizations of Sobolev inequalities on metric spaces

J. Math. Anal. Appl. 344 (2008) 1093–1104

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J. Math. Anal. Appl. www.elsevier.com/locate/jmaa

Characterizations of Sobolev inequalities on metric spaces Juha Kinnunen, Riikka Korte ∗ Institute of Mathematics, PO Box 1100, FI-02015, Helsinki University of Technology, Finland

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 September 2007 Available online 3 April 2008 Submitted by J. Xiao

We discuss Maz’ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces. © 2008 Elsevier Inc. All rights reserved.

Keywords: Sobolev inequality Isocapacitary inequality Metric spaces

1. Introduction There is a well-known connection between the isoperimetric and Sobolev inequalities, see for example [5,6,16]. Indeed, by the isoperimetric inequality, we have

| E |(n−1)/n  c (n)Hn−1 (∂ E ),

(1.1)

where E is a smooth enough subset of Rn . Here | E | is the Lebesgue measure and Hn−1 ( E ) denotes the (n − 1)-dimensional Hausdorff measure. The constant c (n) can be chosen so that (1.1) becomes an equality when E is a ball. The Sobolev inequality states that



(n−1)/n  |u |n/(n−1) dx  c (n) |∇ u | dx

Rn

(1.2)

Rn

for every u ∈ C 0∞ (Rn ). The smallest constant in (1.2) is the same as the constant in (1.1). The Sobolev inequality follows from the isoperimetric inequality through the co-area formula. On the other hand, the isoperimetric inequality can be deduced from the Sobolev inequality. When the gradient is integrable to a power which is greater than one, the isoperimetric inequality has to be replaced with an isocapacitary inequality. When the exponent is one, capacity and Hausdorff content are equivalent and hence it does not matter which one we choose. In this case, it is enough to have the isocapacitary inequality for balls instead of all sets. Hence isocapacitary and Sobolev inequalities are different aspects of the same phenomenon. This elegant approach to Sobolev inequalities is due to Maz’ya, see [13–17]. In particular, our work is closely related to [14], where similar questions are studied in a more general context of topological spaces. In weighted Euclidean spaces, these characterizations have been studied in [25]. It also known that Maz’ya type characterizations of Sobolev inequalities are available on Riemannian manifolds. The purpose of our work is to emphasize the fact that Maz’ya type characterizations give necessary and sufficient conditions for Sobolev type inequalities in the context of Newtonian spaces introduced in [23]. In the borderline case when

*

Corresponding author. E-mail addresses: juha.kinnunen@tkk.fi (J. Kinnunen), [email protected].fi (R. Korte).

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2008.03.070

© 2008 Elsevier Inc.

All rights reserved.

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J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

the gradient is integrable to power one, we apply equivalence of the capacity of order one and the Hausdorff content of co-dimension one. This is based on metric space versions of the co-area formula and boxing inequality, see [10,12,18,19]. Rather standard assumptions in analysis on metric measure spaces include a doubling condition for the measure and validity of some kind of Sobolev–Poincaré inequality. Despite the fact that plenty of analysis has been done in this general context, very little is known about the basic assumptions. Several necessary conditions are known, but unfortunately only few sufficient conditions are available so far, see [22]. On Riemannian manifolds, Grigor’yan and Saloff-Coste observed that the doubling condition and the Poincaré inequality are not only sufficient but also necessary conditions for a scale invariant parabolic Harnack principle for the heat equation, see [7,20,21]. The advantage of Maz’ya type characterizations is that they apply in a more general context than Riemannian manifolds. 2. Preliminaries We assume that X = ( X , d, μ) is a metric measure space equipped with a metric d and a Borel regular outer measure μ such that 0 < μ( B ) < ∞ for all balls B = B (x, r ) = { y ∈ X: d(x, y ) < r }. In what follows, Ω stands for an open bounded subset of X unless otherwise stated. The measure μ is said to be doubling if there exists a constant c D  1, called the doubling constant, such that









μ B (x, 2r )  c D μ B (x, r )

for all x ∈ X and r > 0. In this paper, a path in X is a rectifiable nonconstant continuous mapping from a compact interval to X . A path can thus be parameterized by arc length. By saying that a condition holds for p-almost every path with 1  p < ∞, we mean that it fails only for a path family with zero p-modulus. A family Γ of curves is of zero p-modulus if there is a nonnegative Borel measurable function ρ ∈ L p ( X ) such that for all curves γ ∈ Γ , the path integral γ ρ ds is infinite. A nonnegative Borel function g on X is an upper gradient of an extended real valued function u on X if for all paths γ joining points x and y in X we have

  u (x) − u ( y ) 



g ds,

(2.1)

γ



whenever both u (x) and u ( y ) are finite, and γ g ds = ∞ otherwise. If g is a nonnegative measurable function on X and if (2.1) holds for p-almost every path, then g is a p-weak upper gradient of u. Let 1  p < ∞. If u is a function that is integrable to power p in X , let

  u  N 1, p ( X ) =

 1/ p

 p

p

|u | dμ + inf

g dμ

g

X

,

X

where the infimum is taken over all p-weak upper gradients of u. The Newtonian space on X is the quotient space





N 1, p ( X ) = u: u  N 1, p ( X ) < ∞ /∼, where u ∼ v if and only if u − v  N 1, p ( X ) = 0. For properties of Newtonian spaces, we refer to [23]. Let E be a subset of Ω . We write u ∈ A( E , Ω) if u | E = 1 and u | X \Ω = 0. Definition 2.1. Let E ⊂ Ω . The p-capacity of E with respect to Ω is



cap p ( E , Ω) = inf

p

g u dμ, Ω

where the infimum is taken over all continuous functions u ∈ A( E , Ω) with p-weak upper gradients g u . If there are no such functions, then cap p ( E , Ω) = ∞. We say that a property regarding points in X holds p-quasieverywhere (p-q.e.) if the set of points for which the property does not hold has capacity zero. To be able to compare the boundary values of Newtonian functions, we need a Newtonian space with zero boundary values. Let E be a measurable subset of X . The Newtonian space with zero boundary values is the space 1, p





N 0 ( E ) = u | E : u ∈ N 1, p ( X ) and u = 0 p-q.e. on X \ E . 1, p

1, p

Note that if cap p ( X \ E , X ) = 0, then N 0 ( E ) = N 1, p ( X ). The space N 0 ( E ) equipped with the norm inherited from N 1, p ( X ) is a Banach space, see Theorem 4.4 in [24].

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

Definition 2.2. We say that X supports a weak (1, p )-Poincaré inequality if there exist constants c p > 0 and for all balls B (x, r ) of X , all integrable functions u on X and all upper gradients g u of u, we have







– |u − u B (x,r ) | dμ  c p r

τ  1 such that

 1/ p (2.2)

,

B (x,τ r )

B (x,r )

where

p

g u dμ

–

1095

 u B = – u dμ =



1

u dμ.

μ( B )

B

B

3. Functions with zero boundary values In this section, we give necessary and sufficient conditions for Sobolev inequalities of type



1/q |u |q dν



 1/ p

p

 cS

g u dμ

Ω

,

Ω 1, p

where the constant c S is independent of u ∈ N 0 (Ω) ∩ C (Ω), and μ and ν are Borel regular outer measures. We consider two ranges of indices separately. We have chosen to study continuous Newtonian functions, but our arguments do not depend on the choice of the function space. For example, similar results also hold for all Newtonian functions and Lipschitz functions if the definition of p-capacity is adjusted accordingly. Remark 3.1. In the standard versions of Poincaré inequality, the inequality depends on the diameter of the set. Therefore the constant c S also depends strongly on the diameter of the set in many cases. 3.1. The case 1  p  q < ∞ Let u : Ω → [−∞, ∞] be a

∞

 |u | p dμ = p

μ-measurable function. By the well-known Cavalieri principle

λ p −1 μ( E λ ) dλ,

0

Ω

where









E λ = x ∈ Ω : u (x) > λ . The following simple integral inequality will be useful for us. Notice that the equality occurs when p = q. Lemma 3.2. If u : Ω → [−∞, ∞] is μ-measurable and 0 < p  q < ∞, then



 ∞

1/q q

|u | dμ

 p

1/ p λ

p −1

μ( E λ )

p /q

dλ

.

0

Ω

Proof. We have



 p q/ p dμ |u |

 p /q

X

 |u | p f dμ,

= sup

 f s 1

X

as where s = q/(q − p ) is the Hölder conjugate of q/ p. By  f s we denote the L s (μ)-norm of f . Define a measure μ 

( A ) = μ

| f | dμ A

for every

μ-measurable set A ⊂ X . If μ( E ) > 0 and  f s  1, we conclude that 

( E ) = μ E

| f | dμ  μ( E )1−1/s



 1/ s | f |s dμ

E

 μ( E )1−1/s = μ( E ) p /q .

(3.1)

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Here we used the Hölder inequality. Hence



∞



= p |u | p dμ

|u | p f dμ  X

X

( E λ ) dλ  p λ p −1 μ

0

∞

λ p −1 μ( E λ ) p /q dλ.

0

Taking supremum over all functions f with  f s  1 completes the proof.

2

Next we prove a strong type inequality for the capacity. When p = 1 the obtained estimate reduces to the co-area formula. The proof is based on a general truncation argument, see page 110 in [16]. A similar argument has been used for example in [1–3,8,9,20]. 1, p

Lemma 3.3. Let u ∈ N 0 (Ω) ∩ C (Ω) and 1  p < ∞. Then

∞

λ p −1 cap p ( E λ , Ω) dλ  22p −1

0



p

g u dμ, Ω

where g u is a p-weak upper gradient of u. Proof. A straightforward calculation shows that

∞ λ

p −1

2 j ∞ 

cap p ( E λ , Ω) dλ =

j =−∞

0

∞ 



λ p −1 cap p ( E λ , Ω) dλ

2 j −1





2 j − 2 j −1 2 j ( p −1) cap p ( E 2 j−1 , Ω)

j =−∞

∞ 1 

=

2

2 jp cap p ( E 2 j−1 , Ω)

j =−∞

= 2 p −1

∞ 

2 jp cap p ( E 2 j , Ω).

j =−∞

Let uj =

⎧ ⎨ 1, ⎩

1− j

2

0,

if u  2 j ,

|u | − 1, if 2 j −1 < u < 2 j , if u  2 j −1 .

Then u j ∈ A( E 2 j , Ω). This implies that



cap p ( E 2 j , Ω)  2 p (1− j )

p

g u dμ E 2 j −1 \ E 2 j

and consequently ∞ 

∞ 

2 jp cap p ( E 2 j , Ω) 

j =−∞

The claim follows from this.

j =−∞

2 jp + p (1− j )



p



g u dμ  2 p E 2 j −1 \ E 2 j

p

g u dμ. Ω

2

The following result gives a necessary and sufficient condition for a Sobolev inequality in terms of an isocapacitary inequality. This is a metric space version of a corollary on page 113 of [16]. Remark 3.4. We do not need the doubling condition in this theorem.

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

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Theorem 3.5. Suppose that 1  p  q < ∞. (i) If there is a constant γ such that

ν ( E ) p/q  γ cap p ( E , Ω)

(3.2)

for every E ⊂ Ω , then



1/q |u |q dν

  cS

 1/ p

p

g u dμ

Ω

(3.3)

Ω 1, p

for every u ∈ N 0 (Ω) ∩ C (Ω) with c S depending only on γ and p. 1, p

(ii) If (3.3) holds for every u ∈ N 0 (Ω) ∩ C (Ω) and if the constant c S is independent of u, then (3.2) holds for every E ⊂ Ω with γ = cS . Proof. (i) By Lemma 3.2, (3.2) and Lemma 3.3, we obtain



 ∞

1/q q

|u | dν

 p

1/ p λ

p −1

ν(Eλ)

p /q

  γp

dλ

0

Ω

1/ p

∞ λ

p −1

cap p ( E λ , Ω) dλ



 2p −1 1/ p



 γ p2

0

p gu

 1/ p dμ

.

Ω

(ii) If u ∈ A( E , Ω) is continuous, then by (3.3), we have



ν ( E )1/q 

1/q |u |q dν

  cS

Ω

p

 1/ p

g u dμ

.

Ω

The claim follows by taking the infimum on the right-hand side.

2

Remark 3.6. The previous theorem gives a necessary and sufficient condition for the Hardy inequality

  Ω

|u (x)| dist(x, X \ Ω)

p

 dμ  c H

p

g u dμ,

(3.4)

Ω 1, p

where the constant c H is independent of u ∈ N 0 (Ω) ∩ C (Ω). Indeed, (3.4) holds if and only if

 E

1 dist(x, X \ Ω) p

dμ  γ cap p ( E , Ω)

for every E ⊂ Ω . Thus Theorem 3.5 is a generalization of Theorem 4.1 in [11]. In the metric space context, the Hardy inequality has also been studied in [4]. 3.2. The case p = 1 When p = 1 and Ω = X , the isocapacitary inequalities reduce to isoperimetric inequalities. Moreover, in this case we can improve Theorem 3.5 under the additional assumptions that the measure is doubling and the space supports a Poincaré inequality. Indeed, it is enough that condition (3.2) is satisfied for all balls. To prove that, we will need equivalence of the capacity of order one and the Hausdorff content of co-dimension one



H∞ ( K ) = inf h

∞  μ( B (xi , ri )) i =1

ri

: K⊂

∞ 

 B ( xi , r i ) .

i =1

Theorem 3.7. Let X be a complete metric space with a doubling measure μ. Suppose that X supports a weak (1, 1)-Poincaré inequality. Let K be a compact subset of X . Then 1 c

cap1 ( K )  Hh∞ ( K )  c cap1 ( K ),

where c depends only on the doubling constant and the constants in the weak (1, 1)-Poincaré inequality. The proof is based on co-area formula and a metric space version of so–called boxing inequality. For more details, see [10]. A similar result has been studied in Mäkäläinen [19].

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Theorem 3.8. Let X be a complete metric space with a doubling measure μ. Suppose that X supports a weak (1, 1)-Poincaré inequality. Suppose that 1  q < ∞. If there is a constant γ such that

ν ( B )1/q  γ cap1 ( B , X )

(3.5)

for every ball B ⊂ X , then



1/q |u |q dν

  cS

X

g u dμ,

(3.6)

X

where c S is independent of u ∈ N 1,1 ( X ) ∩ C 0 ( X ). Proof. First we prove that if the space satisfies (3.5) for all balls in X , then it satisfies the same condition for all compact sets with a different constant. Let K ⊂ X be compact, ε > 0 and { B (xi , r i )}∞ be a covering of K such that i =1

Hh∞ ( K ) 

∞  μ( B (xi , ri ))

ri

i =1

− ε.

Since q  1, we have

ν ( K )1/q 

∞  

1/q

ν B ( xi , r i )

.

i =1

Because









u i (x) = 1 − dist x, B (xi , r i ) /r i + belongs to A( B (xi , r i ), X ), and g i = χ B (xi ,2ri ) /r i is an upper gradient of u i , we have







cap1 B (xi , r i ) 

g i dμ  c D

μ( B (xi , ri )) ri

X

.

By combining the above estimates and (3.5), we conclude

ν ( K )1/q 

∞  

1/q

ν B ( xi , r i )

γ

i =1

∞ 





cap1 B (xi , r i )  γ c D

i =1

∞  μ( B (xi , ri )) i =1

ri

   γ c D Hh∞ ( K ) + ε .

The claim follows by Theorem 3.7 as ε → 0. Now the theorem follows as in the proof of Theorem 3.5. Note that since u has compact support and is continuous, we can as well consider compact level sets {|u |  t } instead of open sets. 2 3.3. The case 1  q < p < ∞ In the case 1  q < p < ∞, the isocapacitary inequality takes a different form. Let E j , j = − N , − N + 1, . . . , N , N + 1, be such that E j ⊂ Ω and E j ⊂ E j +1 for j = − N , − N + 1, . . . , N. We define



γ = sup

N   j =− N

q/( p −q) ( p −q)/q

ν ( E j ) p/q

,

cap p ( E j , E j +1 )

(3.7)

where the supremum is taken over all sequences of sets as above. The following result is a metric space version of a theorem on page 120 of [16]. Theorem 3.9. Suppose that 1  q < p < ∞. (i) If γ < ∞, then



1/q |u |q dν

  cS

Ω

p

 1/ p

g u dμ

,

Ω 1, p

where c S is independent of u ∈ N 0 (Ω) ∩ C (Ω). (ii) If (3.8) holds for every u ∈

1, p N 0 (Ω) ∩

C (Ω) and if the constant c S is independent of u, then γ < ∞.

(3.8)

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

1099

Proof. (i) We have



2 j+1

∞ 

q

|u | dν =

j =−∞

Ω

∞ 

λq−1 ν ( E λ ) dλ  q

q

2( j +1)(q−1) 2 j ν ( E 2 j ) = q2q−1

j =−∞

2j

∞ 

2 jq ν ( E 2 j ).

j =−∞

By the Hölder inequality, ∞ 

∞  

2 jq ν ( E 2 j ) =

j =−∞

∞  



j =−∞

Let uj =

⎧ ⎪ ⎨ 1, ⎪ ⎩

q/ p



2 jp cap p ( E 2 j , E 2 j−1 )

cap p ( E 2 j , E 2 j−1 )

j =−∞



q/ p

ν ( E 2 j ) p/q

q/( p −q) ( p −q)/ p   ∞

ν ( E 2 j ) p/q cap p ( E 2 j , E 2 j−1 )

q/ p 2

jp

cap p ( E 2 j , E 2 j−1 )

.

j =−∞

if |u | > 2 j ,

|u |−2 j −1 2 j −1

, if 2 j −1 < |u |  2 j , if |u |  2 j −1 .

0,

Then





g u j dμ  2−( j −1) p p

cap p ( E 2 j , E 2 j−1 ) 

p

g u dμ. E 2 j −1 \ E 2 j

Ω

It follows that ∞ 

2 jp cap p ( E 2 j , E 2 j−1 )  2 p

j =−∞



∞ 



p

g u dμ = 2 p

j =−∞ E

2 j −1

\E2 j

p

g u dμ Ω

and consequently





q/ p

p

|u |q dν  c

g u dμ

Ω

.

Ω

(ii) Let E j be as in the statement of the theorem, and define

λj =

N   i= j

1/( p −q)

ν(Ei )

j = − N , − N + 1, . . . , N ,

,

cap p ( E i , E i +1 )

and λ N +1 = 0. Let u j ∈ A( E j , E j +1 ) be continuous, and define

⎧ ⎨ (λ j − λ j +1 )u j + λ j +1 in E j +1 \ E j , u = λ− N in E − N , ⎩ 0 in Ω \ E N +1 . 1, p

Then u ∈ N 0 (Ω) ∩ C (Ω). By the Cavalieri principle



∞ q

|u | dν = q

q−1

λ

ν ( E λ ) dλ =

λ j q

λq−1 ν ( E λ ) dλ 

j =− N λ j +1

0

Ω

N 

N  j =− N





ν ( E j ) λqj − λqj+1 .

From this we conclude that



N 

p /q

ν ( E j )(λ j − λ j+1 )q

 

j =− N

N  j =− N

 cS

N  j =− N



ν ( E j ) λqj − λqj+1

p /q 

 Ω

 (λ j − λ j +1 ) p

p

g u j dμ. E j +1 \ E j

 p /q |u |q dν



  cS

p

g u dν = c S Ω

N  j =− NE



p

g u dμ j +1 \ E j

1100

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

Taking the infimum on the right-hand side, we arrive at



p /q

N 

q

ν ( E j )(λ j − λ j+1 )

 cS

j =− N

 λ j − λ j +1 = N  

1/( p −q)

ν(E j )

,

cap p ( E j , E j +1 )

we obtain

j =− N

(λ j − λ j +1 ) p cap p ( E j , E j +1 ).

j =− N

Since



N 

q/( p −q)  p /q

ν ( E j ) p/q cap p ( E j , E j +1 )

 cS

N   j =− N

ν ( E j ) p/q

q/( p −q)

cap p ( E j , E j +1 )

,

2

and the claim follows.

Next we present an integral version of Theorem 3.9. See also page 30 in [17] for the Euclidean case. Theorem 3.10. Let q < p and μ(Ω) < ∞, and



λ p (s) = inf cap p (G ): G ⊂ Ω and ν (G )  s .

Then ν(Ω)

t p /q

q/( p −q)

λ p (t )

0

dt

 cI < ∞

t

(3.9) 1, p

if and only if the Sobolev inequality (3.8) holds for every u ∈ N 0 (Ω) ∩ C (Ω) with a constant c S that is independent of u. Proof. First, assume that (3.9) holds. Let s = p /q and s = p /( p − q) be the Hölder conjugate of s. Then



∞ 

u q dν  2q

2 jq ν



2 j < u < 2 j +1



j =−∞

Ω



∞ 

q

2

2

jp

cap p



j =−∞



j =−∞

q/ p  ν(Ω)

p

c

1/ s  q/ p  ∞  (ν ({u > 2 j }) − ν ({u > 2 j +1 }))s

 u>2  cap p ({u > 2 j })s /s j

g u dμ Ω 1/ s 

 cI



dt

q/ p

p

g u dμ

c

λ p (t )s /s

0



t

1/ s 

s −1

.

Ω

Here we used the Hölder inequality, monotonicity of λ p (t ) and the fact that cap p



u > 2j



 2− jp + p



p

g u dμ.

{2 j −1
Assume now that (3.8) holds. For every j ∈ Z, let u j ∈ N 0 (Ω) ∩ C (Ω) be a function such that 0  u j  1, and



 

p

g u j dμ  λ p 2 j + ε j , Ω

with 0  ε j  λ p (2 j ). Let u = sup β j u j , j

where

 βj =

2j

λ p (2 j )

1/( p −q) .

ν ({u j = 1})  2 j

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

Now

 u q dν  Ω

and



p

g u dμ 

2

q

βj 2j

(3.10)

j =−∞



  p      p β j λp 2 j + ε j  2 β j λp 2 j .

j

Ω

As

∞ 1 

1101

(3.11)

j

  q p β j 2 j = β j λp 2 j ,

it follows by (3.8), (3.10) and (3.11) that

 (2 j ) p /( p −q)  q = β j 2 j  c, j q /( p − q ) λ p (2 ) j

j

and ν(Ω)

t p /q

q/( p −q)

dt

λ p (t )

0

t

c

2

by monotonicity of λ p .

4. Functions with general boundary values In this section, we obtain necessary and sufficient conditions for Sobolev–Poincaré inequalities of type



1/q

|u − a|q dν

inf

a∈R



 1/ p

p

c

g u dμ

Ω

,

Ω

where u ∈ N 1, p (Ω) ∩ C (Ω) and 1  p  q < ∞. To this end, we shall need the concept of conductivity, see Chapter 4 in [16]. Let Ω ⊂ X be a bounded open set. Let F be a closed subset of Ω and let G be an open subset of Ω such that F ⊂ G. The open set C = G \ F is called a conductor and



B( F , G , Ω) = u ∈ N 1, p (Ω) ∩ C (Ω): u  1 in F and u  0 in Ω \ G

is the set of admissible functions. The number



con p ( F , G , Ω) =

p

g u dμ

inf

u ∈B ( F ,G ,Ω)

Ω

is called the p-conductivity of C . The next result can be proved in the same way as Lemma 3.3. Lemma 4.1. Let G ⊂ Ω be open and 1  p < ∞. Suppose that u ∈ N 1, p (Ω) ∩ C (Ω) such that u = 0 in Ω \ G. Then

∞



λ p −1 con p ( E λ , G , Ω) dλ  22p −1

0

p

g u dμ. Ω

The following result is a metric space version of a theorem on page 210 of [16]. Theorem 4.2. Suppose that 1  p  q < ∞ and let G ⊂ Ω be open. (i) If there is a constant γ such that

ν ( F ) p/q  γ con p ( F , G , Ω) for every F ⊂ G, then



1/q

|u |q dν Ω

 c

p

(4.1)

 1/ p

g u dμ Ω

for every u ∈ N 1, p (Ω) ∩ C (Ω) such that u = 0 in Ω \ G. Here the constant c is independent of u. (ii) If (4.2) holds for every u ∈ N 1, p (Ω) ∩ C (Ω) such that u = 0 in Ω \ G, then (4.1) holds for every F ⊂ G with γ = c.

(4.2)

1102

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

Proof. (i) We conclude by Lemma 3.2, (4.1) and Lemma 4.1 that



 ∞

1/q q

|u | dν

 p

1/ p λ

p −1

ν(Eλ)

p /q

  pγ

dλ

0

Ω

1/ p

∞ λ

p −1

con p ( E λ , G , Ω) dλ

0

  1/ p  p  p γ 22p −1 g u dμ . Ω

(ii) If u ∈ B ( F , G , Ω), then inequality (4.2) implies that

1/q



ν ( F )1/q 



|u |q dν

c

 1/ p

p

g u dμ

Ω

.

Ω

2

The claim follows by taking the infimum on the right-hand side.

Theorem 4.3. Suppose that 1  p  q < ∞ and that Ω ⊂ X is a bounded open set. (i) If there is a constant γ such that

ν ( F ) p/q  γ con p ( F , G , Ω)

(4.3)

for every conductor G \ F with ν (G )  ν (Ω)/2, then



1/q |u − a|q dν

inf

a∈R

 c

 1/ p

p

g u dμ

Ω

(4.4)

Ω

for every u ∈ N 1, p (Ω) ∩ C (Ω). (ii) If (4.4) holds for every u ∈ N 1, p (Ω) ∩ C (Ω), then (4.3) holds for every conductor G \ F with ν (G )  ν (Ω)/2. Proof. (i) Let

α ∈ R be such that



ν x ∈ Ω : u (x)  α



1



ν (Ω)

2

and



ν x ∈ Ω : u (x) > α



1



ν (Ω).

2

Now (u − α )+ ∈ N 1, p (Ω) ∩ C (Ω) and u = 0 in Ω \ G, where





G = x ∈ Ω : u (x) > α . Clearly

ν (G )  12 ν (Ω) and by (4.3),

ν ( F ) p/q  γ con p ( F , G , Ω) for every F ⊂ G. By Theorem 4.2, we have



1/q

q

(u − α )+ dν





 1/ p

p

c

g u dμ

.

{x∈Ω : u (x)>α }

Ω

Similarly,



q

1/q

(α − u )+ dν





 1/ p

p

c

g u dμ

.

{x∈Ω : u (x)<α }

Ω

A combination of these estimates implies that





1/q |u − α |q dμ



Ω

Ω



p gu

{x∈Ω : u (x)>α }

c

p

g u dμ Ω

+  1/ p dμ

1/q

q

(u − α )+ dμ Ω



c 



1/q

q

(α − u )+ dμ





+c {x∈Ω : u (x)<α }

 1/ p .

p

g u dμ

 1/ p

J. Kinnunen, R. Korte / J. Math. Anal. Appl. 344 (2008) 1093–1104

(ii) Let G \ F be a conductor with u = 1 on F . Since



ν (G )  12 ν (Ω) and suppose that u ∈ N 1, p (Ω) ∩ C (Ω) such that u = 0 in Ω \ G and



1/q

|u − u Ω |q dν Ω

1/q

|u − a|q dν

 Ω

+ |a − u Ω |ν (Ω)1/q 1/q



|u − a|q dν

 Ω

 + ν (Ω)1/q – |u − a| dν Ω

1/q



|u − a|q dν



 + ν (Ω)1/q

Ω



1/q

– |u − a|q dν Ω

1/q |u − a|q dν

=2

1103

,

Ω

we have



1/q |u − a|q dν

inf

a∈R

1/q

 |u − u Ω |q dν



Ω

 a∈R

Ω

1/q |u − a|q dν

 2 inf

.

Ω

Now

 c

p gu

q/ p



Ω

 |u − u Ω |q dν =



dμ

|u − u Ω |q dν + |u Ω |q ν (Ω \ G ). G

Ω 1 2

ν (Ω \ G )  ν (Ω) imply that  q/ p p |u Ω |q ν (Ω)  c g u dμ .

This and the fact that

Ω

Since



1/q |u |q dν

Ω

we have

1/q

  |u Ω |ν (Ω)1/q +

|u − u Ω |q dν Ω



ν ( F )1/q 

1/q |u |q dν

Ω

 c

p

 c

q

g u dμ

 1/ p ,

Ω

 1/ p

g u dμ Ω

and by taking an infimum over all functions u, we have

ν ( F ) p/q  c con p ( F , G , Ω).

2

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