A SIMPLE PROOF OF HARDY’S INEQUALITYIN A LIMITING CASE
FUTOSHI TAKAHASHI
Abstract. In this short note, we provide a simple proof of Hardy’s
inequality in a limiting case. In the proof we do not need any rearrangement technique or the one-dimensional argument.

1. Introduction
Let Ω be a bounded domain in RN with 0 ∈ Ω. The classical Hardy’s
inequality is of the formµ¶ ZZ
N −p p|u|pdx ≤|∇u|p dx
(1.1)
pp|x|
ΩΩ
for all u ∈ W01,p (Ω), where N ≥ 3 and 1 < p < N . See [3] for its simple
proof which uses the Fundamental Theorem of Calculus³´pand Hölder’s inN −pequality only. It is well known that the constantis optimal andp
never attained in W01,p (Ω).
For p = N , the inequality (1.1) loses its sense and instead of (1.1) theinequality¶ ZµZ|u(x)|NN −1 N|∇u|N dx
(1.2)³
´N dx ≤N
Ω |x|N log ReΩ|x|holds for all u ∈ W01,N (Ω), where R = supx∈Ω |x|. Again, the constant¡ N −1 ¢Nis known to be optimal; see for example, [1], [2]. We call (1.2)
N
as Hardy’s inequality in a limiting case. Main aim of this short note isto provide a simple proof of Hardy’s inequality in a limiting case. We do
not need any rearrangement technique such as Polya-Szegö inequality forthe spherical decreasing rearrangement, or a technical one-dimensional argument. Also our method can provide the sharper inequality treated in [8],[2], [9] and [7].

Date: August 25, 2014.
1991 Mathematics Subject Classification. Primary 35A23, Secondary 26D15, 46E35.
12F. TAKAHASHI

Theorem 1.1. Let Ω be a bounded domain in RN , N ≥ 2 with 0 ∈ Ω. Let
g : (1, +∞) → R be a C 2 function with the properties thatg 0 (s) < 0,g 00 (s) > 0

for any s > 1(1.3)

for any s > 1.

(1.4)

and there exists C > 0 such that(−g 0 (s))2(N −1)(g 00 (s))N −1≤C

Put R = supx∈Ω |x|. Then the inequalityµµµ¶ Z¶¶N −2 µ¶N −1 NRe|u(x)|NRe000−g log
g logdxNN|x||x|Ω |x|³³´´2(N −1)¯¯
Z −g 0 log Re¯|x|x ¯¯N¯dx≤³ ³´´N −1 ¯∇u ·|x| ¯Ωg 00 log Re|x|(1.5)

holds true for any u ∈ W01,N (Ω).
Corollary 1.2. Let Ω be a bounded domain in RN , N ≥ 2 with 0 ∈ Ω.
Then the inequality¯µ¶ ZZ ¯¯N −1 N|u(x)|Nx ¯¯N¯(1.6)´N dx ≤³¯∇u · |x| ¯ dx
NΩ |x|N log RΩ|x|holds for all u ∈ W01,N (Ω), where R = supx∈Ω |x|.
Note that, different from the function
function

1”N“
R|x|N log |x|1”N appeared in (1.2), the“|x|N log Re|x|in (1.6) becomes unbounded when |x| ∼ 0 and also|x| ∼ R.
Corollary 1.3. Let Ω be a bounded domain in RN , N ≥ 2 with 0 ∈ Ω. Let
α > 0. Then the inequality
¯¯µ¶
ZZ
¯
N −1 N N|u(x)|Nx ¯¯Nα(N −1) ¯dx ≤|x|α(1.7)¯∇u · |x| ¯ dx
N −α(N −1)NΩ |x|Ωholds for all u ∈ W01,N (Ω).
If we take α = NN−1 in (1.7), we have¯¯µ¶ZZZ¯x ¯¯NNNNN¯dx ≤ R|∇u| dx ,|u(x)| dx ≤|x| ¯∇u ·|x| ¯ΩΩΩthus (1.7) can be seen as a generalization of Poincaré’s inequality for u ∈W01,N (Ω).

HARDY INEQUALITY

3

The proof of Theorem 1.1 relies on the divergence theorem and Hölder’sinequality. Similar “simple” approaches have been proposed in [10], [4], and[5], mainly to derive (1.1). Their proof uses the identityµdivx|x|λ¶

N −λ|x|λ=for |x| =6 0, λ ∈ R.

On the other hand, the identitydiv ³x

R|x|N log |x|N −1³´N
R|x|N log |x|´N −1  =for |x| 6= 0, R,

will be the base of our proof. See the next section.
Another approach to the limiting case of Hardy’s inequality, using themean integral of a Schwarz symmetrization, has been done by Ioku; see[6]:Remark 1.4.

2. Proof of Theorem.
In this section, we prove Theorem 1.1. For ε > 0 small, put¡
¢1/2Rε = sup |x|2 + 2ε2,Xε (x) = logx∈ΩRε e(|x|2 + ε2 )1/2andψε (x) = g(Xε (x)) ∈ C 2 (Ω).

(2.1)

We calculate¡¢N −1|∇ψε (x)|N −2 ∇ψε (x) = −g 0 (Xε )Ã¡¢∆N ψε (x) = div |∇ψε (x)|N −2 ∇ψε (x)= (N − 1)(−g 0 (Xε ))N −2 g 00 (Xε )|x|N −2 x(|x|2 + ε2 )N −1|x|N2 N(|x|2 + ε )!
,+ (−g 0 (Xε ))N −12(N − 1)ε2 |x|N −2(|x|2 + ε2 )Nwhere we have used the assumption that g 0 (s) < 0. For u ∈ W01,N (Ω),
divergence theorem assures thatZZ¡¢N|u| ∆N ψε dx = − ∇ |u|N · |∇ψε |N −2 ∇ψε dx.
(2.2)
ΩΩ

,

4F. TAKAHASHIThe RHS of (2.2) is estimated from above as¯ Z¯¯¯N −2N −2¯|RHS of (2.2)| = ¯N|u|u∇u · |∇ψε |∇ψε dx¯¯Ω¯ ZÃ! ¯¯¯N −2 x · ∇u|x|¯¯= ¯N|u|N −2 u(−g 0 (Xε ))N −1dx¯
N−1¯¯(|x|2 + ε2 )ΩÃ!
Z|x|N −2 |x · ∇u|≤N|u|N −1 (−g 0 (Xε ))N −1dx(|x|2 + ε2 )N −1Ω! N −1
ÃZN¢N −2 00|u|N |x|N ¡ 0−g (Xε )g (Xε )dx
×≤NN
Ω (|x|2 + ε2 )ÃZ¯
¯N ! N1
¯¯x
¯ dx
×
,
(−g 0 (Xε ))2(N −1) (g 00 (Xε ))−(N −1) ¯¯∇u ·|x| ¯Ωwhere we have used Hölder’s inequality. On the other hand, the LHS of(2.2) is estimated from below as|LHS of (2.2)|(Z= (N − 1) |u|N (−g 0 (Xε ))N −2 g 00 (Xε )ΩZ≥ (N − 1)ΩThus we have(N − 1)

|u|N (−g 0 (Xε ))N −2 g 00 (Xε )Z|u|N |x|NNΩ (|x|2 + ε2 )ÃZ
≤N|u|N |x|N|x|N(|x|2 + ε2 )N

×
Ω

dx.

(−g 0 (Xε ))N −2 g 00 (Xε )dx! N −1N0N −2 00(−g (Xε ))NΩ (|x|2 + ε2 )ÃZ2N −20
N −1 2ε |x|+(−g(X))
ε(|x|2 + ε2 )N(|x|2 + ε2 )N|x|Ng (Xε )dx

×¯N ! N1¯¯¯x¯ dx(−g 0 (Xε ))2(N −1) (g 00 (Xε ))−(N −1) ¯¯∇u ·,|x| ¯

which implies
¶ Z
µ|x|N
N −1 N|u|N (−g 0 (Xε ))N −2 g 00 (Xε )dx
N(|x|2 + ε2 )NΩ¯¯Z(−g 0 (Xε ))2(N −1) ¯¯x ¯¯N≤¯∇u · |x| ¯ dx.
N −1Ω (g 00 (Xε ))Finally, we let ε → 0 in thesides of the above inequality. Note that³ both´
ReRε → R and Xε → log |x| a.e. x ∈ Ω as ε → 0. By using Fatou’s)dx

HARDY INEQUALITY

5lemma in the LHS and the Lebesgue dominated convergence theorem withthe assumption (1.4) in the RHS, we conclude (1.5) holds.
Proof of Corollary 1.2, 1.3.
Proof. For the proof of Corollary 1.2, let g(s) = − log(s − 1) for s > 1. Then1
1
we see g 0 (s) = − s−1< 0, g 00 (s) = (s−1)
2 > 0, and(−g 0 (s))2(N −1)(g 00 (s))N −1=1( (s−1))2(N −1)
1
N −1( (s−1)
2)=1

for any s > 1.

Thus the assumptions (1.3),(1.4) are satisfied. Insertingµ¶¶µRe−11
Re000
g log
==, g log
RR 2|x||x|log |x|(log |x|)
into (1.5), we obtain (1.6).
For Corollary 1.3, take g(s) = e−αs for s > 1. We easily see that(−g 0 (s))2(N −1)(g 00 (s))N −1= e−(N −1)αs ≤ 1

for any s > 1and¶µ ¶α¶µ ¶αµµ|x||x|ReRe
200= −α=αg log, g log
.
|x|
Re|x|
Re
Using these and (1.5), we have (1.7).
0Acknowledgement. Part of this work was done when the author visited
KAIST and KIAS in Korea, August, 2014. He thanks both institutes for its
support and hospitality. The author was also supported by JSPS Grant-inAid for Scientific Research (B), No. 23340038, and JSPS Grant-in-Aid for
Challenging Exploratory Research, No. 24654043.
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F. TAKAHASHI

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Department of Mathematics, Osaka City University & OCAMI, Sumiyoshiku, Osaka, 558-8585, Japan
E-mail address: futoshi@sci.osaka-cu.ac.jp