Characterizations of Sobolev inequalities on metric spaces

0.19 MB·English

by  Juha Kinnunen

CHARACTERIZATIONS OF SOBOLEV INEQUALITIES
ON METRIC SPACES
JUHA KINNUNEN AND RIIKKA KORTE
Abstract. WepresentisocapacitarycharacterizationsofSobolev
8 inequalities in very general metric measure spaces.
0
0
2

n
a 1. Introduction
J

4
1 ThereisawellknownconnectionbetweentheisoperimetricandSobolev
inequalities. By the isoperimetric inequality, we have
]
P
|E|(n−1)/n ≤ c(n)Hn−1(∂E), (1.1)
A
h. where E is a smooth enough subset of Rn, |E| is the Lebesgue measure
at and Hn−1 is the (n−1)-dimensional Hausdorff measure. The constant
m c(n) is chosen so that (1.1) becomes an equality when E is a ball. The
[ Sobolev inequality states that

2
(n−1)/n
v
|u|n/(n−1)dx ≤ c(n) |∇u|dx (1.2)
7
9 (cid:18)ZRn (cid:19) ZRn
0 for every u ∈ C∞(Rn). The smallest constant in (1.2) is the same as
1 0
. the constant in (1.1). The Sobolev inequality follows from the isoperi-
9
0 metric inequality through the co-area formula. On the other hand, the
7 isoperimetric inequality can be deduced from the Sobolev inequality,
0
see for example [5]. This shows that the isoperimetric and Sobolev
:
v
inequalities are different aspects of the same phenomenon. Originally
i
X this observation is due to Federer and Fleming [6] and Maz′ya [11].
r
a 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 due to the boxing inequality, it is enough to have the isoca-
pacitary inequality for balls instead of all sets. This elegant approach
to Sobolev inequalities is due to Maz′ya, see [12] and [13]. Usually this
characterization leads to descriptions of the best possible constants in
Sobolev inequalities. However, the aim of the present work is not so
much to study best possible constants but rather study necessary and
2000 Mathematics Subject Classification. 46E35, 31C45.
1
2 JUHA KINNUNENAND RIIKKAKORTE
sufficient conditions for Sobolev–Poincar´e inequalities in a general met-
ricspacecontext. InweightedEuclideanspaces, thesecharacterizations
have been studied in [20].
Rather standard assumptions in analysis on metric measure spaces in-
clude a doubling condition for the measure and validity of some kind
of Sobolev–Poincar´e inequality. Despite the fact that plenty of anal-
ysis 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. On
Riemannian manifolds, Grigor’yan and Saloff-Coste observed that the
doubling condition and the Poincar´e inequality are not only sufficient
but also necessary conditions for a scale invariant parabolic Harnack
principle for the heat equation, see [15], [16] and [7]. It also known that
Maz′ya type characterizations of Sobolev inequalities are available on
Riemannian manifolds. The purpose of this work is to show that this
is also the case on a very general metric measure spaces.
2. Preliminaries
We assume that X = (X,d,µ) is a metric measure space equipped with
ametricdandaBorelregularoutermeasureµsuchthat0 < 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 ≥ 1,
D
called the doubling constant, such that
µ(B(x,2r)) ≤ c µ(B(x,r))
D
for all x ∈ X and r > 0.
In this paper, a path in X is a rectifiable non-constant continuous map-
ping 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 non-negative Borel
measurable function ρ ∈ Lp(X) such that for all curves γ ∈ Γ, the path
integral ρds is infinite.
γ
A nonnegative Borel function g on X is an upper gradient of an ex-
R
tended real valued function u on X if for all paths γ joining points x
and y in X we have
|u(x)−u(y)| ≤ gds, (2.1)
Zγ
CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 3
whenever both u(x) and u(y) are finite, and gds = ∞ otherwise.
γ
If g is a nonnegative measurable function on X and if (2.1) holds for
R
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
1/p
kuk = |u|pdµ+inf gpdµ ,
N1,p(X)
g
(cid:16)ZX ZX (cid:17)
where the infimum is taken over all p-weak upper gradients of u. The
Newtonian space on X is the quotient space
N1,p(X) = {u : kuk < ∞}/∼,
N1,p(X)
where u ∼ v if and only if ku − vk = 0. For properties of
N1,p(X)
Newtonian spaces, we refer to [18].
Let E beasubset of Ω. Wewriteu ∈ A(E,Ω) ifu = 1andu = 0.
|E |X\Ω
Definition 2.1. Let E ⊂ Ω. The p–capacity of E with respect to Ω is
cap (E,Ω) = inf gpdµ,
p u
ZΩ
where the infimum is taken over all continuous functions u ∈ A(E,Ω)
with p–weak upper gradients g . If there are no such functions, then
u
cap (E,Ω) = ∞.
p
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
N1,p(E) = {u| : u ∈ N1,p(X) and u = 0 p–q.e. on X \E}.
0 E
Note that if cap (X \E,X) = 0, then N1,p(E) = N1,p(X). The space
p 0
N1,p(E) equipped with the norm inherited from N1,p(X) is a Banach
0
space, see Theorem 4.4 in [19].
Definition 2.2. We say that X supports a weak (1,p)–Poincar´e in-
equality if there exist constants c > 0 and τ ≥ 1 such that for all balls
p
B(x,r) of X, all integrable functions u on X and all upper gradients
g of u, we have
u
1/p
|u−u |dµ ≤ c r gpdµ , (2.2)
B(x,r) p u
ZB(x,r) (cid:18)ZB(x,τr) (cid:19)
where
1
u = udµ = udµ.
B
µ(B)
ZB ZB
4 JUHA KINNUNENAND RIIKKAKORTE
3. Functions with zero boundary values
In this section, we give necessary and sufficient conditions for Sobolev
inequalities of type
1/q 1/p
|u|qdν ≤ c gpdµ ,
S u
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
where the constant c is independent of u ∈ N1,p(Ω)∩C(Ω), and µ and
S 0
ν are Borel regular outer measures. We consider two ranges of indices
separately.
We have chosen to study continuous Newtonian functions, but our ar-
guments do not depend on the choice of the function space. For exam-
ple, 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 Poincar´e inequality, the in-
equality 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 µ–
measurable function. By the well-known Cavalieri principle
∞
|u|pdµ = p λp−1µ(E )dλ,
λ
ZΩ Z0
where
E = {x ∈ Ω : |u(x)| > λ}.
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 ∞ 1/p
|u|qdµ ≤ p λp−1µ(E )p/qdλ . (3.1)
λ
(cid:18)ZΩ (cid:19) (cid:18) Z0 (cid:19)
Proof. We have
p/q
(|u|p)q/p dµ = sup |u|pf dµ,
(cid:18)ZX (cid:19) kfks≤1ZX
where s = q/(q−p) is the H¨older conjugate of q/p. By kfk we denote
s
the Ls(µ)–norm of f. Define a measure µ as
µ(A) = |f|edµ
ZA
e
CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 5
for every µ–measurable set A ⊂ X. If µ(E) > 0 and kfk ≤ 1, we
s
conclude that
1/s
µ(E) = |f|dµ ≤ µ(E)1−1/s |f|sdµ
ZE (cid:18)ZE (cid:19)
≤ µ(E)1−1/s = µ(E)p/q.
e
Here we used the H¨older inequality. Hence
∞
|u|pf dµ ≤ |u|pdµ = p λp−1µ(E )dλ
λ
ZX ZX Z0
∞
≤ p λp−e1µ(E )p/qdλ. e
λ
Z0
Taking supremum over all functions f with kfk ≤ 1 completes the
s
(cid:3)
proof.
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 [12]. A similar
argument has been used for example in [3], [15] and [8].
Lemma 3.3. Let u ∈ N1,p(Ω)∩C(Ω) and 1 ≤ p < ∞. Then
0
∞
λp−1cap (E ,Ω)dλ ≤ 22p−1 gpdµ,
p λ u
Z0 ZΩ
where g is a p-weak upper gradient of u.
u
Proof. A straightforward calculation shows that
∞
λp−1cap (E ,Ω)dλ
p λ
Z0
∞ 2j
= λp−1cap (E ,Ω)dλ
p λ
j=−∞Z2j−1
X
∞
≤ (2j −2j−1)2j(p−1)capp(E2j−1,Ω)
j=−∞
X
∞
1
= 2 2jpcapp(E2j−1,Ω)
j=−∞
X
∞
= 2p−1 2jpcap (E ,Ω).
p 2j
j=−∞
X
Let
1, if u ≥ 2j,
u = 21−j|u|−1, if 2j−1 < u < 2j,
j 
0, if u ≤ 2j−1.


6 JUHA KINNUNENAND RIIKKAKORTE
Then u ∈ A(E ,Ω). This implies that
j 2j
cap (E ,Ω) ≤ 2p(1−j) gpdµ
p 2j u
ZE2j−1\E2j
and consequently
∞ ∞
2jpcap (E ,Ω) ≤ 2jp+p(1−j) gpdµ
p 2j u
j=−∞ j=−∞ ZE2j−1\E2j
X X
≤ 2p gpdµ.
u
ZΩ
(cid:3)
The claim follows from this.
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 [12].
Remark 3.4. We do not need the doubling condition in this theorem.
Theorem 3.5. Suppose that 1 ≤ p ≤ q < ∞.
(i) If there is a constant γ such that
ν(E)p/q ≤ γcap (E,Ω) (3.2)
p
for every E ⊂ Ω, then
1/q 1/p
|u|qdν ≤ c gpdµ (3.3)
S u
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
for every u ∈ N1,p(Ω)∩C(Ω) with c depending only on γ and
0 S
p.
(ii) If (3.3) holds for every u ∈ N1,p(Ω)∩C(Ω) and if the constant
0
c is independent of u, then (3.2) holds for every E ⊂ Ω with
S
γ = c .
S
Proof. (i) By Lemma 3.2, (3.2) and Lemma 3.3, we obtain
1/q ∞ 1/p
|u|qdν ≤ p λp−1ν(E )p/qdλ
λ
(cid:18)ZΩ (cid:19) (cid:18) Z0 (cid:19)
∞ 1/p
≤ γp λp−1cap (E ,Ω)dλ
p λ
(cid:18) Z0 (cid:19)
1/p
≤ γp22p−1 1/p gpdµ .
u
(cid:18)ZΩ (cid:19)
(cid:0) (cid:1)
(ii) If u ∈ A(E,Ω) is continuous, then by (3.3), we have
1/q 1/p
ν(E)1/q ≤ |u|qdν ≤ c gpdµ .
S u
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
(cid:3)
The claim follows by taking the infimum on the right-hand side.
CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 7
Remark 3.6. The previous theorem gives a necessary and sufficient
condition for the Hardy inequality
p
|u(x)|
dµ ≤ c gpdµ, (3.4)
dist(x,X \Ω) H u
ZΩ(cid:18) (cid:19) ZΩ
where the constant c is independent of u ∈ N1,p(Ω)∩C(Ω). Indeed,
H 0
(3.4) holds if and only if
1
dµ ≤ γcap (E,Ω)
dist(x,X \Ω)p p
ZE
for every E ⊂ Ω. Thus Theorem 3.5 is a generalization of Theorem
4.1 in [10]. 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 Poincar´e 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
∞ ∞
µ(B(x ,r ))
Hh (K) = inf i i : K ⊂ B(x ,r ) .
∞ r i i
( i )
i=1 i=1
X [
Theorem 3.7. Let X be a complete metric space with a doubling mea-
sure µ. Suppose that X supports a weak (1,1)–Poincar´e inequality. Let
K be a compact subset of X. Then
1
cap (K) ≤ Hh (K) ≤ ccap (K),
c 1 ∞ 1
where c depends only on the doubling constant and the constants in the
weak (1,1)–Poincar´e inequality.
The proof is based on co-area formula and a metric space version of
so–called boxing inequality. For more details, see [9]. A similar result
has been studied in M¨ak¨al¨ainen [14].
Theorem 3.8. Let X be a complete metric space with a doubling mea-
sure µ. Suppose that X supports a weak (1,1)–Poincar´e inequality.
Suppose that 1 ≤ q < ∞. If there is a constant γ such that
ν(B)1/q ≤ γcap (B,X) (3.5)
1
for every ball B ⊂ X, then
1/q
|u|qdν ≤ c g dµ, (3.6)
S u
(cid:18)ZX (cid:19) ZX
where c is independent of u ∈ N1,1(X)∩C (X).
S 0
8 JUHA KINNUNENAND RIIKKAKORTE
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(x ,r )}∞ be a covering of K
i i i=1
such that
∞
µ(B(x ,r ))
Hh (K) ≥ i i −ε.
∞ r
i
i=1
X
Since q ≥ 1, we have
∞
ν(K)1/q ≤ ν(B(x ,r ))1/q.
i i
i=1
X
Because
u (x) = (1−dist(x,B(x ,r ))/r )
i i i i +
belongs to A(B(x ,r ),X), and g = χ /r is an upper gradient
i i i B(xi,2ri) i
of u , we have
i
µ(B(x ,r ))
i i
cap (B(x ,r )) ≤ g dµ ≤ c .
1 i i i D r
ZX i
By combining the above estimates and (3.5), we conclude
∞
ν(K)1/q ≤ ν(B(x ,r ))1/q
i i
i=1
X
∞
≤ γ cap (B(x ,r ))
1 i i
i=1
X
∞
µ(B(x ,r ))
i i
≤ γc
D
r
i
i=1
X
≤ γc (Hh (K)+ε).
D ∞
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. (cid:3)
3.3. The case 1 ≤ q < p < ∞. In the case 1 ≤ q < p < ∞, the
isocapacitary inequality takes a different form. Let E , j = −N,−N +
j
1,...,N,N+1 be such that E ⊂ Ω and E ⊂ E for j = −N,−N+
j j j+1
1,...,N. We define
(p−q)/q
N ν(E )p/q q/(p−q)
j
γ = sup , (3.7)
cap (E ,E )
"j=−N(cid:18) p j j+1 (cid:19) #
X
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
[12].
CHARACTERIZATIONS OF SOBOLEV INEQUALITIES 9
Theorem 3.9. Suppose that 1 ≤ q < p < ∞.
(i) If γ < ∞, then
1/q 1/p
|u|qdν ≤ c gpdµ , (3.8)
S u
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
where c is independent of u ∈ N1,p(Ω)∩C(Ω).
S 0
(ii) If (3.8) holds for every u ∈ N1,p(Ω)∩C(Ω) and if the constant
0
c is independent of u, then γ < ∞.
S
Proof. (i) We have
∞ 2j+1
|u|qdν = q λq−1ν(E )dλ
λ
ZΩ j=−∞ Z2j
X
∞
≤q 2(j+1)(q−1)2jν(E )
2j
j=−∞
X
∞
=q2q−1 2jqν(E ).
2j
j=−∞
X
By the H¨older inequality,
∞
2jqν(E )
2j
j=−∞
X
∞ ν(E )p/q q/p
=j=−∞(cid:18)capp(E22jj,E2j−1)(cid:19) 2jpcapp(E2j,E2j−1) q/p
X (cid:0) (cid:1)
∞ ν(E )p/q q/(p−q) (p−q)/p
≤ 2j
j=−∞(cid:18)capp(E2j,E2j−1)(cid:19) !
X
∞ q/p
× 2jpcapp(E2j,E2j−1) .
!
j=−∞
X
Let
1, if |u| > 2j,
|u|−2j−1
uj =  , if 2j−1 < |u| ≤ 2j,
 2j−1

0, if |u| ≤ 2j−1.

Then 

capp(E2j,E2j−1) ≤ gupj dµ ≤ 2−(j−1)p gupdµ
ZΩ ZE2j−1\E2j
10 JUHA KINNUNENAND RIIKKAKORTE
It follows that
∞ ∞
2jpcapp(E2j,E2j−1) ≤ 2p gupdµ
j=−∞ j=−∞ZE2j−1\E2j
X X
= 2p gpdµ
u
ZΩ
and consequently
q/p
|u|qdν ≤ c gpdµ .
u
ZΩ (cid:18)ZΩ (cid:19)
(ii) Let E be as as in the statement of the theorem, and define
j
N 1/(p−q)
ν(E )
i
λ = , j = −N,−N +1,...,N,
j
cap (E ,E )
i=j (cid:18) p i i+1 (cid:19)
X
and λ = 0. Let u ∈ A(E ,E ) be continuous, and define
N+1 j j j+1
(λ −λ )u +λ in E \E ,
j j+1 j j+1 j+1 j
u = λ in E ,
 −N −N
0 in Ω\E .
N+1
Then u ∈ N1,p(Ω)∩C(Ω). By the Cavalieri principle
0
∞ N λj
|u|qdν =q λq−1ν(E )dλ = q λq−1ν(E )dλ
λ λ
ZΩ Z0 j=−N Zλj+1
X
N
≥ ν(E )(λq −λq ).
j j j+1
j=−N
X
From this we conclude that
p/q
N p/q N
ν(E )(λ −λ )q ≤ ν(E )(λq −λq )
j j j+1 j j j+1
!
(cid:18)j=−N (cid:19) j=−N
X X
p/q N
≤ |u|qdν ≤ c gpdν = c gpdµ
S u S u
(cid:18)ZΩ (cid:19) ZΩ j=−NZEj+1\Ej
X
N
≤ c (λ −λ )p gp dµ.
S j j+1 uj
j=−N ZEj+1\Ej
X
Taking the infimum on the right-hand side, we arrive at
p/q
N N
ν(E )(λ −λ )q ≤ c (λ −λ )pcap (E ,E ).
j j j+1 S j j+1 p j j+1
!
j=−N j=−N
X X