HARDY'S INEQUALITY FOR W -FUNCTIONS ON RIEMANNIAN ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 9, Pages 2745–2754 S 0002-9939(99)04849-2 Article electronically published on April 23, 1999

HARDY’S INEQUALITY FOR W01,p -FUNCTIONS ON RIEMANNIAN MANIFOLDS VLADIMIR M. MIKLYUKOV AND MATTI K. VUORINEN (Communicated by Christopher D. Sogge) Abstract. We prove that for every Riemannian manifold X with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class W01,p (X ). We also study manifolds satisfying Hardy’s inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold.

1. Main results Let X be a connected noncompact Riemannian C 2 -manifold without boundary. We denote by ρ(x0 , x00 ) the distance between two points x0 , x00 ∈ X . For an arbitrary point x ∈ X we set ε(x) = inf lim inf ρ(x, yk ), (yk )

where the infimum is taken over all sequences of points (yk ) ⊂ X , which do not have points of accumulation in X . Let α(t), β(t) : [0, ∞) → [0, ∞) be positive continuous functions. We fix constants p, q such that 1 < p ≤ q < ∞. Let Ω ⊂ X be an arbitrary domain, and let ∂Ω be its boundary. We consider a weighted volume Z VX (Ω) = α(ε(x))q dv. Ω

Here, dv is the element of volume on the manifold X . Below we shall assume that IX = VX (X ) ≤ ∞. We denote a weighted area Z AX (∂Ω) = β(ε(x))α(ε(x))(p−1)q/p dH n−1 , ∂Ω

where dH is the element of the (n − 1)-dimensional Hausdorff measure on ∂Ω. We consider isoperimetric profiles of the Riemannian manifold X with weight functions α and β. An isoperimetric profile of the manifold X is the function n−1

(1.1)

θX : [0, IX ) → R+ ,

θ(0) = 0,

Received by the editors May 20, 1997 and, in revised form, November 24, 1997. 1991 Mathematics Subject Classification. Primary 53C21. c

1999 American Mathematical Society

2745

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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VLADIMIR M. MIKLYUKOV AND MATTI K. VUORINEN

defined by θX (τ ) = inf{AX (∂Ω) : Ω ⊂ Y

a compact domain

(∂Ω) < ∞, VX (Ω) = τ }, with H i.e. the isoperimetric profile θX is the best function among the functions θ satisfying θ VX (Ω) ≤ AX (∂Ω). (1.2) n−1

In the special case of surfaces this definition goes back to Ahlfors [1, p.188]. In the general case see, for example, [4]. In general, the isoperimetric profile θX (τ ) is difficult to compute. It is also difficult to estimate the isoperimetric profile in terms of curvature and other geometric data. We give some of these cases below. 1.3. Example. Let X be a complete, simply connected, n−dimensional Riemannian manifold with nonpositive sectional curvature. We write VX (Ω) = vol(Ω)

and

AX (∂Ω) = area(∂Ω).

By [12], [7] for every n there is a constant cn > 0 such that (n−1)/n cn vol(Ω) ≤ area(∂Ω) and it follows that

θX (τ ) ≥ cn τ (n−1)/n .

1.4. Example. Let X be a complete, simply connected Riemannian manifold, dim X = n. If the sectional curvature KX of X satisfies KX ≤ k < 0, k = const, then p (n − 1) (−k) vol(Ω) ≤ area(∂Ω) ([19], p.504; [6], 34.2.6) and thus p θX (τ ) ≥ (n − 1) (−k) τ. Furthermore, we shall consider only isoperimetric profiles θ = θX (t) of a special type. In fact, we shall assume that I (p−1)/p ZX dt  B = sup r1/q  (1.5) < ∞. θ(t)p/(p−1) r∈(0,IX ) r

Below by functions of the class W01,p (X ), p > 1, we shall understand the closure of the class of the C0∞ (X )-functions in the norm kukW 1,p = kukp + k|∇u|kp where k.kp stands for the standard norm in Lp (X ). We have the following theorem. 1.6. Theorem. Let X be an n−dimensional connected noncompact Riemannian C 2 -manifold without boundary. If the manifold X has an isoperimetric profile with the properties (1.1), (1.2) and (1.5), then for an arbitrary function u ∈ W01,p (X ), p > 1, we have  1/q  1/p Z Z  |α(ε(x))u(x)|q dv  (1.7) ≤ λ  (β(ε(x))|∇u(x)|)p dv  , X

X


HARDY’S INEQUALITY FOR W01,p -FUNCTIONS

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where q ≥ p is arbitrary and λ is such that B ≤ λ ≤ B q 1/q (

(1.8)

q (p−1)/p ) . q−1

In the case when X = D is a proper subdomain of Rn , the function ε(x) = ρ(x, ∂D) is the distance from a point x ∈ D to the boundary ∂D, the inequality (1.7) yields a generalized Hardy inequality. We record a widely known variant of it (α(t) = 1/t, β(t) = 1 and p = q) Z Z u(x) p (1.9) dv ≤ λp |∇u(x)|p dv, u ∈ W01,p (D), p > 1. ρ(x, ∂D) D

D

A one-dimensional version of Hardy’s classical inequality can be found in [10, p. 175]; for its generalization to the case of domains D ⊂ Rn see [2], [15], [14], [17], [18], and [13]. For α(t) = β(t) ≡ 1 and p = q the inequality (1.7) can be written as the wellknown Poincar´e-Sobolev inequality (cf. [19], [11, p. 48], and the literature cited there) Z Z (1.10) |u(x)|p dv ≤ λp |∇u(x)|p dv, u ∈ W01,p (D), p > 1. D

D

1.11. Example. Let X = D be a domain in R2 , and let p = q = 2, β ≡ 1. We use the metric ds2 = α2 (ρ(x, ∂D)) (dx21 + dx22 ), and consider the manifold X = (D, ds). We have for an arbitrary domain Ω ⊂ D with H 1 (∂Ω) < ∞ : Z Z V(D, ds) (Ω) = α(ρ(x, ∂D))2 dx1 dx2 , A(D, ds) (∂Ω) = α(ρ(x, ∂Ω)) dH 1 . Ω

∂Ω

The Gaussian curvature K of X = (D, ds) is equal (see, for example, [8, §13 of Chapter 2]) 1 ∆ log α(ρ). K =− 2 α (ρ) It is nonpositive if and only if ∆ log α ρ(x, ∂D) ≥ 0, ∀ x ∈ D. Since |∇ρ(x, ∂D)| ≡ 1, this inequality is equivalent to the following inequality 0 0 α0 (ρ) α (ρ) ∆ρ + (1.12) ≥ 0, ∀ x ∈ D. α(ρ) α(ρ) Therefore, if the condition (1.12) is true then we have for the isoperimetric profile of the manifold (D, ds) (see Example 1.3) θ(D, ds) (τ ) ≥ c2 τ 1/2 , and the property (1.5) holds. Consequently, we obtain for such domains Z Z 2 α ρ(x, ∂D) u2 (x) dx ≤ λ2 |∇u(x)|2 dx, D

D


u ∈ W01,2 (D).

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In the special case α(τ ) = property

1 τ,

we can conclude that if the domain D ⊂ R2 has the ∆ρ(x, ∂D) ≤

1 , ρ(x, ∂D)

then the Hardy’s inequality (1.9) (with p = 2) is correct for this domain. For convex domains the function ρ(x, ∂D) is superharmonic ([3]) and hence the above inequality holds. The next theorem characterizes manifolds, on which a generalized Hardy’s inequality is valid. 1.13. Theorem. Let X be an n-dimensional connected noncompact Riemannian C 2 -manifold without boundary, satisfying the condition (1.14)

{x ∈ X : ε(x) ≥ t} ⊂⊂ X ,

∀t ∈ (0, ε0 ),

x∈X

If for every function u ∈ W01,p (X ), constant λ < ∞, then (1.15)

 sup  r∈(0,ε0 )

Zε0

ε0 = sup ε(x).

p > 1, the inequality (1.7) holds with the

1/q  r Z  α(t)q S(t)dt 0

r

where

1 β(t)p S(t)

1/(p−1)

(p−1)/p dt

≤ λ,

Z dH n−1 ,

S(t) =

t ∈ (0, ε0 ),

ε(x)=t

is the (n − 1)-dimensional Hausdorff measure of the set {x ∈ X : ε(x) = t}. A relation of the type of (1.15) contains important information on the structure of of the manifold X . For the purpose of illustration we give two particular cases of Theorem 1.13. In the case α(t) = 1/t, β(t) ≡ 1, p = q, we have: 1.16. Corollary. Let X be an n−dimensional connected noncompact Riemannian C 2 -manifold without boundary, satisfying the condition (1.14). If for all functions u ∈ W01,p (X ), p > 1, the inequality (1.9) holds, then (1.17)

1 p/(p−1) lim sup log r r→0 0

Zr 0

1 1/(p−1) dt < ∞. S(t)

and for each pair of numbers r , r , 0 < r0 < r00 < ε0 , , Z Z λp dv dv (1.18) ≤ . 00 p p ε(x) ε(x) (log rr0 )p ε(x)>r 00

00

r 0 r

ε(x) 0 everywhere in the set {x ∈ X : |u(x)| > 0} except possibly at finitely many points. This can always be achieved on the expense of a small perturbation of the function u (see, for example [8, Theorem 5, p.488]). We set Et = {x ∈ X : |u(x)| = t}, Xt = {x ∈ X : |u(x)| > t}. On the basis of the Kronrod-Federer co-area formula we can write Z Z∞ Z dH n−1 q . I(t) ≡ α(ε(x)) dv = dτ α(ε(x))q |∇u| Xt

t

Eτ

Therefore for almost all t ≥ 0 we have Z dH n−1 . I 0 (t) = − α(ε(x))q |∇u| Et

Using similar arguments we have Z Z |α(ε(x))u(x)|q dv = X

0

Z∞ =−

(2.1)

q 0

∞

Z τ q dτ ZIX τ q (I) dI,

τ I (τ )dτ = 0

α(ε(x))q dH n−1 Eτ

0

where τ : [0, IX ) → [0, ∞) is the inverse function of I(τ ). Furthermore, we have p  Z  β(ε(x))α(ε(x))(p−1)q/p dH n−1  Et

 ≤

Z

Et

p−1  Z n−1 dH  β(ε(x))p |∇u|p−1 dH n−1   α(ε(x))q , |∇u| Et


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and therefore

Z β(ε(x))p |∇u|p−1 dH n−1 Et

 p Z 1 ≥ 0 p−1  β(ε)α(ε(x))(p−1)q/p dH n−1  . |I (t)|

(2.2)

Et

Because u has a compact support in X , the open set Xt ⊂⊂ X . On the basis of the isoperimetric relation (1.2) we can write Z θ(I(t)) ≤ β(ε)α(ε(x))q(p−1)/p dH n−1 . Et

Thus we get by (2.2) θ(I(t))p ≤ |I 0 (t)|p−1

(2.3)

Z β(ε(x))p |∇u|p−1 dH n−1 . Et

Now we can transform the integral on the right side of (1.7). Applying once again the Kronrod-Federer co-area formula, we have Z

p

β(ε(x)) |∇u|

Z∞ dv =

Z Eτ

0

X

β(ε(x))p |∇u|p−1 dH n−1 .

dτ

Therefore the inequality (2.3) yields Z (2.4) X

β(ε(x)) |∇u|

p

Z∞ dv ≥ 0

θ(I(τ ))p dτ = |I 0 (τ )|p−1

ZIX

θ(I)p |τ 0 (I)|p dI.

0

Combining the relations (2.1) and (2.4), we get p 1/p R β(ε(x)) |∇u| dv

X ≥ inf q 1/q R α(ε(x)) |∇u| dv X

IRX 0

!1/p θ(I)p |τ 0 (I)|p dI IRX 0

!1/q

≡

1 , λ

τ (I)p dI

where the infimum is taken over all nonincreasing AC−functions τ with τ (I), τ (IX ) = 0. The one-dimensional functional on the right side of the above inequality has been extensively studied. Applying Theorem 4 from [16, the inequality (13) §1.3.1], we arrive at the dual inequality (1.8), where the quantity B is defined by the relation (1.5). What was said above proves the validity of the inequality (1.7) for C0∞ (X )functions. For the class W01,p (X ) the proof follows after a passage to limit in the Wp1 -norm.


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3. Proofs of Theorem 1.13 and Corollaries The proofs are very simple. The condition (1.14) guarantees the precompactness of the open sets Xt = {x ∈ X : ε(x) > t},

t ∈ (0, ε0 ).

Let ϕ(t) be an arbitrary AC-function on [0, ε0 ) which vanishes for all t ∈ [0, t0 ) for some t0 > 0. Each of the functions ϕ∗ (x) = ϕ(ε(x)) has a support contained in one of the sets Xt and is therefore a member of the class W01,p (X ). Choosing in the inequality (1.7) the function u(x) = ϕ∗ (x), we obtain 1/q  1/p  Z Z p  |α(ε(x))φ(ε(x))|q dv  β(ε(x))|φ0 (ε(x))| |∇ε(x)| dv  . ≤λ X

X

Integrating along the level sets of ε(x), we get 1/q  Zε0 Z n−1 dH   q q   α(τ ) φ(τ ) dτ |∇ε(x)| 0

ε(x)=τ

 Zε0  ≤ λ  β(τ )p |φ0 (τ )|p dτ 0

Z

1/p  |∇ε(x)|p−1 dH n−1 

.

ε(x)=τ

Because ε(x) is the intrinsic distance function on X we have |∇ε(x)| ≡ 1 almost everywhere. Thus we get for 1 < p ≤ q < ∞ (3.1)

1/q ε 1/p ε Z0 Z0  α(τ )q S(τ )φ(τ )q dτ  ≤ λ  β(τ )p S(τ )|φ0 (τ )|p dτ  , 0

0

valid for every AC−function ϕ(t) on (0, ε0 ) vanishing close to 0. This relation between one-dimensional integrals has been extensively studied in the literature, see e.g. [16, p. 40]. The validity of the inequality (3.1) for all ϕ with the given properties implies the validity of the inequality (1.15). We next prove Corollary 1.16. We choose in (1.15) the weight functions α(t) = 1/t, β(t) ≡ 1 and p = q. We arrive at the estimate 1/p  r (p−1)/p ε Z0 Z 1/(p−1) S(t)  1  (3.2) dt dt ≤ λ. sup  p t S(t) r∈(0,ε0 ) 0

r

However, by H¨ older’s inequality we get for all r0 , r00 , 0 < r0 < r00 < ε0 , (3.3)

 r00 p  r00   r00 p−1 Z Z Z 1 S(t) r00 p  dt  )1/(p−1) dt = ≤ dt  ( . log 0 r t tp S(t) r0

r0

r0


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Let r0 < r00 < ε0 . Then we have from (3.2) r00 (log 0 )p r Zε0 ≤

r00 p log 0 r

Zr

1 1/(p−1) p−1 dt ≤ S(t)

r0

Zr

00

r0

S(t) dt tp

0

Zr 1 p−1 S(t) 1/(p−1) p dt ≤ λ / dt tp S(t) 0

r0

and

Zr00

0

0

, Zr00

1 1/(p−1) dt S(t)

r0

1 1/(p−1) p−1 dt ≤ λp . S(t)

Therefore, 00 r R

log

r00 −p 0 r0

1/(p−1) 1 dt S(t) Rr0 0

1 S(t)

−

Rr 0

0

1/(p−1) 1 dt S(t) p−1

1/(p−1)

≥ λ−p

dt

and further

(log

R r00

1 1/(p−1) dt r −p/(p−1) 0 ( S(t) ) − 1 ≥ λ−p/(p−1) ) 0 R 0 r 1 1/(p−1) r dt 0 ( S(t) ) 00

or, equivalently, (3.4)

Z 0

r

00

00

p 1 1/(p−1) r ) ( dt ≥ [1 + λ− p−1 (log 0 )p/(p−1) ] S(t) r

0

Z

r

0

( 0

1 1/(p−1) ) dt. S(t)

00

The inequality (3.4) holds for all r < r . Passing to the limit with r0 → 0, we arrive at (1.17). Further from (3.2) and (3.3) , Zr00 Zε0 r00 p S(t) S(t) (3.5) dt dt ≤ λp . log 0 p r t tp r 00

r0

We observe that Zr r0

and

00

S(t) dt = tp

Zr

00

r0

Zε0 r 00

dt tp

Z

dH n−1 = |∇ε|

ε(x)=t

S(t) dt = tp

Z ε(x)>r 00

Z r 0