Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 75, pp. 1–12.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES

NGUYEN TUAN DUY, HUY BAC NGUYEN

Communicated by Jesus Ildefonso Diaz

Abstract. We study some versions of the cylindrical Hardy identities and

inequalities in the style of Badiale-Tarantello [2]. We show that the best con-
stants of the cylindrical Hardy inequalities can be improved when we consider

functions on half-spaces.

1. Introduction

The main subject of this note is the celebrated Hardy inequality on RN , N ≥ 3:
for u ∈ C∞0 (RN ): ∫

RN
|∇u|2 dx ≥

(N − 2

2

)2 ∫
RN

|u|2

|x|2
dx (1.1)

with optimal constant (N−2
2 )2. Because of their important roles in many areas of

mathematics, the Hardy type inequalities have been well-studied and there is a
vast literature. See the monographs [3, 25, 28, 29, 40], for instance, that are typical
references on the topic.

It is well-known that (N−2
2 )2 in (1.1) is never achieved by nontrivial functions.

Therefore, many efforts have been devoted to enhance the Hardy inequalities. One
way to do so is to add extra nonnegative terms to the right-hand side of (1.1). The
first result in this direction was established in [8] where Brezis and Vázquez proved

that for u ∈W 1,2
0 (Ω). Ω is a bounded domain in RN , N ≥ 3, with 0 ∈ Ω, it holds∫

Ω

|∇u|2 dx ≥
(N − 2

2

)2 ∫
Ω

|u|2

|x|2
dx+ z2

0ω
2/N
N |Ω|−2/N

∫
Ω

|u|2 dx. (1.2)

Here ωN is the volume of the unit ball and z0 = 2.4048 . . . is the first zero of the

Bessel function J0(z). The constant z2
0ω

2/N
N |Ω|−2/N is optimal when Ω is a ball.

However, z2
0ω

2
N

N |Ω|−2/N is not attained in W 1,2
0 (Ω). Hence, Brezis and Vázquez also

conjectured that z2
0ω

2/N
N |Ω|−2/N

∫
Ω
|u|2 dx is just a first term of an infinite series of

extra terms that can be added to the right-hand side of (1.2). This question was
investigated by many authors. We refer the interested reader to [1, 4, 9, 10, 11, 12,
18, 21, 22, 23, 26, 37, 45, 46], to name just a few.

Ghoussoub and Moradifam [24, 25] proved the following result to improve, extend
and unify several results about the Hardy type inequalities:

2010 Mathematics Subject Classification. 26D10, 35A23, 46E35.

Key words and phrases. Cylindrical Hardy inequality; Bessel pair.
c©2020 Texas State University.

Submitted May 24, 2020. Published July 16, 2020.

1



2 N. T. DUY, H. B. NGUYEN EJDE-2020/75

Theorem 1.1. Let 0 < R ≤ ∞, V and W be positive C1-functions on (0, R)

such that
∫ R

0
1

rN−1V (r)
dr = ∞ and

∫ R
0
rN−1V (r)dr < ∞. Then the following two

statements are equivalent: (1) (rN−1V, rN−1c1W ) is a Bessel pair on (0, R) for
some c1 > 0. (2) ∫

BR

V (|x|)|∇u|2 dx ≥ c2
∫
BR

W (|x|)|u|2 dx

for all u ∈ C∞0 (BR) and some c2 > 0.

Here we say that a couple of C1-functions (V,W ) is a Bessel pair on (0, R) if the
ordinary differential equation

y′′(r) +
Vr(r)

V (r)
y′(r) +

W (r)

V (r)
y(r) = 0

has a positive solution on the interval (0, R). See the book [25] for more properties
and examples about the Bessel pair.

Another line of research on the improvements of the Hardy type inequalities is to
replace the usual ∇ by R := x

|x| · ∇. It can be noted that Ru is the radial gradient

of u. Indeed, in the polar coordinate, |Ru| = |∂ru(rσ)| while

|∇u| =
(
|∂ru(rσ)|2 +

|∇SN−1u(rσ)|2

r2

)1/2

.

Actually, the radial derivation plays an important part in the literature. The in-
terested reader is referred to [42] for the roles of the radial derivation R in the
functional and geometric inequalities on homogeneous groups. We also mention
here that the Hardy type inequalities with radial gradient have been intensively
studied recently. See [13, 14, 15, 16, 27, 30, 31, 39, 41, 42, 43], for example.

In an effort to unify many results about the Hardy type inequalities with radial
derivation, and to compute the exact remainders of the Hardy type inequalities,
the authors in [15] have proved the following result.

Theorem 1.2. 0 < R ≤ ∞, V and W be a positive C1-functions on (0, R) such

that
∫ R

0
1

rN−1V (r)
dr =∞ and

∫ R
0
rN−1V (r)dr <∞. Assume that (rN−1V, rN−1W )

is a Bessel pair on (0, R). Then for all u ∈ C∞0 (BR):∫
BR

V (|x|)|Ru|2 dx−
∫
BR

W (|x|)|u|2 dx

=

∫
BR

V (|x|)
∣∣∣R( u

ϕrN−1V,rN−1W ;R

)∣∣∣2ϕ2
rN−1V,rN−1W ;R dx

and ∫
BR

V (|x|)|∇u|2 dx−
∫
BR

W (|x|)|u|2 dx

=

∫
BR

V (|x|)
∣∣∣∇( u

ϕrN−1V,rN−1W ;R

)∣∣∣2ϕ2
rN−1V,rN−1W ;R dx

where ϕrN−1V,rN−1W ;R is the positive solution of

y′′(r) +
(N − 1

r
+
Vr(r)

V (r)

)
y′(r) +

W (r)

V (r)
y(r) = 0

on the interval (0, R).



EJDE-2020/75 CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES 3

In [2], for investigating the existence and nonexistence of cylindrical solutions
for a nonlinear elliptic equation that has been proposed as a model describing
the dynamics of elliptic galaxies, Badiale and Tarantello established the following
cylindrical Hardy type inequality,∫

RN
|∇u(x)|p dx ≥ CN,k,p

∫
RN

|u(x)|p

|y|p
dx (1.3)

where x = (y, z) ∈ Rk × RN−k. The optimal constant CN,k,p = (k−pp )p was also

conjectured in [2] and then verified in [44].
Recently, in [17, 31], the following result about the cylindrical Hardy type in-

equalities with Bessel pairs has been set up.

Theorem 1.3. Let 0 < R ≤ ∞, V and W be positive C1-functions on (0, R).
Assume that (rk−1V, rk−1W ) is a Bessel pair on (0, R). Then for u ∈ C∞0 ({0 <
|y| < R}):∫

0<|y|<R
V (|y|)|∇u(y, z)|2dydz −

∫
0<|y|<R

W (|y|)|u(y, z)|2dydz

=

∫
0<|y|<R

V (|y|)ϕ2(|y|)|∇(
u(y, z)

ϕ(|y|)
)|2dydz.

and ∫
0<|y|<R

V (|y|)| y
|y|
· ∇yu(y, z)|2dydz −

∫
0<|y|<R

W (|y|)|u(y, z)|2dydz

=

∫
0<|y|<R

V (|y|)ϕ2(|y|)| y
|y|
· ∇y(

u(y, z)

ϕ(|y|)
)|2dydz.

Here ϕ is the positive solution of

(rk−1V (r)y′(r))′ + rk−1W (r)y(r) = 0

on the interval (0, R).

Because of their geometric meaning, Hardy’s inequalities have been also studied
extensively on the half-spaces RN+ = {x ∈ RN : x1 > 0}. For instance, Hardy’s
inequalities with distance to the boundary have been investigated in [6, 7, 19, 32, 33],
to name just a few. Improved Hardy type inequalities on half-spaces have also been
set up in, for instance, [5, 34, 35, 36].

It is interesting to note that when one restricts the domain to RN+ , the best
constant of the Hardy inequality can be improved. Indeed, we have the Hardy
inequality on half-space (see, e.g., [24, 38])∫

RN+
|∇u|2 dx ≥

(N
2

)2 ∫
RN+

|u|2

|x|2
dx for u ∈ C∞0 (RN+ ). (1.4)

Here the constant (N/2)2 is optimal. However, if we concern the Hardy inequality
with radial derivation R on RN+ , then it is interesting to note that the best constant
is still ((N − 2)/2)2. Actually, in [32], the authors showed the following identities
to provide a simple interpretation of the aforementioned phenomenon, a direct
understanding of the Hardy inequality on half-spaces (1.4) as well as the “virtual”



4 N. T. DUY, H. B. NGUYEN EJDE-2020/75

ground state in the sense of Frank and Seiringer [20]: for u ∈ C∞0 (RN+ ), it holds∫
RN+
|∇u|2 dx−

(N
2

)2
∫
RN+

|u|2

|x|2
dx =

∫
RN+

∣∣∣∇(|x|N/2 u
x1

)∣∣∣2|x|−Nx2
1 dx,∫

RN+
|Ru|2 dx−

(N − 2

2

)2
∫
RN+

|u|2

|x|2
dx =

∫
RN+

∣∣∣R(|x|N/2 u
x1

)∣∣∣2|x|−Nx2
1 dx.

More generally, the authors in [32] used the factorizations of suitable differential

operators to study a version of Theorem 1.2 on RN+ . Let us denote B
(k)
R the ball

centered at 0 with radius R on Rk. Then we have the following result in [32].

Theorem 1.4. Let 0 < R ≤ ∞, V and W be positive C1-functions on (0, R)

such that
∫ R

0
1

rN+1V (r)
dr = ∞ and

∫ R
0
rN+1V (r)dr < ∞. If (rN+1V, rN+1W ) is a

1-dimensional Bessel pair on (0, R), then for u ∈ C∞0 (RN+ ),∫
B

(N)
R ∩RN+

V (|x|)|∇u|2 dx−
∫
B

(N)
R ∩RN+

[
W (|x|)− V ′(|x|)

|x|
]
|u|2 dx

=

∫
B

(N)
R ∩RN+

V (|x|)
∣∣∣∇( u

ϕrN+1V,rN+1W ;R

1

xN

)∣∣∣2ϕ2
rN+1V,rN+1W ;Rx

2
N dx

and∫
B

(N)
R ∩RN+

V (|x|)|Ru|2 dx−
∫
B

(N)
R ∩RN+

[
W (|x|)− V ′(|x|)

|x|
− (N − 1)

V (|x|)
|x|2

]
|u|2 dx

=

∫
B

(N)
R ∩RN+

V (|x|)
∣∣∣R( 1

ϕrN+1V,rN+1W ;R

u

xN

)∣∣∣2ϕ2
rN+1V,rN+1W ;Rx

2
N dx.

Here ϕrN+1V,rN+1W ;R is the positive solution of

y′′(r) + (
N + 1

r
+
Vr(r)

V (r)
)y′(r) +

W (r)

V (r)
y(r) = 0

on the interval (0, R).

Motivated by the cylindrical Hardy type inequalities studied in [2, 17, 31], and the
Hardy type inequalities on half-spaces in [32], our principal goal of this paper is to
investigate the cylindrical Hardy type inequalities with Bessel pairs and with exact
remainder terms on RN+ . More precisely, let x = (y, z) ∈ Rk × RN−k, 1 ≤ k ≤ N

and y = (x1, w) ∈ R× Rk−1. Our main result reads as follows.

Theorem 1.5. Let 0 < R ≤ ∞, V and W be positive C1-functions on (0, R).
Assume that (rk+1V, rk+1W ) is a Bessel pair on (0, R). Then for u ∈ C∞0 ({0 <
|y| < R} ∩ RN+ ),∫

{0<|y|<R}∩RN+
V (|y|)|∇u|2 dx

−
∫
{0<|y|<R}∩RN+

[
W (|y|)− V ′(|y|)

|y|
]
|u|2 dx

=

∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣∇( u

ϕ(|y|)
1

x1

)∣∣∣2ϕ2(|y|)x2
1 dx



EJDE-2020/75 CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES 5

and ∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣ y|y| · ∇yu∣∣∣2 dx

−
∫
{0<|y|<R}∩RN+

[
W (|y|)− V ′(|y|)

|y|
− (k − 1)

V (|y|)
|y|2

]
|u|2 dx

=

∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣ y|y| · ∇y( u

ϕ(|y|)
1

x1

)∣∣∣2ϕ2(|y|)x2
1 dx.

Here ϕ is the positive solution of

(rk+1V (r)y′(r))′ + rk+1W (r)y(r) = 0

on the interval (0, R).

In Section 4, we will present some cylindrical Hardy type inequalities on half-
spaces as consequences of our main result. Actually, we can obtain as many cylindri-
cal Hardy type inequalities as we can construct Bessel pairs. For several examples
of the Bessel pairs, the interested reader is referred to [25].

2. Some useful calculations

For x̂ = (ŷ, z) ∈ Rk+2 × RN−k, 1 ≤ k ≤ N and ŷ = (t1, t2, t3, w) ∈ R3 × Rk−1,
we denote x = (y, z) ∈ Rk × RN−k, 1 ≤ k ≤ N , and y = (x1, w) ∈ R× Rk−1 where

x1 =
√
t21 + t22 + t23.

Let u ∈ C∞0 (RN+ ). Define a function v : RN+2 → R by v(x̂) = 1
x1
u(x) where

x1 =
√
t21 + t22 + t23. Then v ∈ C∞0 (RN+2). Moreover,

∇RN+2v =
1

x1

( ∂u
∂x1

t1
x1
− 1

x1
u
t1
x1
,
∂u

∂x1

t2
x1
− 1

x1
u
t2
x1
,
∂u

∂x1

t3
x1
− 1

x1
u
t3
x1
,

∂u

∂x2
, . . . ,

∂u

∂xN

)
.

Thus

|∇RN+2v|2 =

N∑
i=2

( 1

x1

)2( ∂u
∂xi

)2

+

3∑
j=1

( 1

x1

∂u

∂x1

tj
x1
− 1

x2
1

u
tj
x1

)2

=

N∑
i=2

(
1

x1
)2
( ∂u
∂xi

)2

+
( 1

x1

)2( ∂u
∂x1

)2

+
|u|2

x4
1

− 2
1

x3
1

u
∂u

∂x1

=
( 1

x1

)2[
|∇RNu|2 +

|u|2

x2
1

− 2
1

x1
u
∂u

∂x1

]
.

(2.1)

and

∆RN+2v =

N∑
i=2

1

x1

∂2u

∂x2
i

+
2

x1

[ 1

x1

∂u

∂x1
− 1

x2
1

u
]

+
1

x1

∂2u

∂x2
1

− 2

x2
1

∂u

∂x1
+

2

x3
1

u

=
1

x1
∆RNu.

(2.2)

Also

∇Rk+2 |ŷ| =
(x1

|y|
t1
x1
,
x1

|y|
t2
x1
,
x1

|y|
t3
x1
,
x2

|y|
, . . . ,

xk
|y|

)
.



6 N. T. DUY, H. B. NGUYEN EJDE-2020/75

Hence

|∇Rk+2 |ŷ|| = |∇Rk |y|| = 1,

∆Rk+2 |ŷ| = k + 1

|y|
=
k + 1

|ŷ|
.

We also have

ŷ

|ŷ|
· ∇ŷv =

k∑
i=2

xi
|y|

1

x1

∂u

∂xi
+

3∑
j=1

tj
|y|

( 1

x1

∂u

∂x1

tj
x1
− 1

x2
1

u
tj
x1

)

=

k∑
i=2

xi
|y|

1

x1

∂u

∂xi
+
x1

|y|

( 1

x1

∂u

∂x1
− 1

x2
1

u
)

=
1

x1

( y
|y|
· ∇yu−

u

|y|

)
.

(2.3)

3. Proof of main results

Proof of Theorem 1.5. For x̂ = (ŷ, z) ∈ Rk+2 × RN−k, 1 ≤ k ≤ N and ŷ =
(t1, t2, t3, w) ∈ R3 × Rk−1, we denote x = (y, z) ∈ Rk × RN−k, 1 ≤ k ≤ N ,

and y = (x1, w) ∈ R×Rk−1 where x1 =
√
t21 + t22 + t23. Let u ∈ C∞0 (RN+ ). Define a

function v : RN+2 → R by v(x̂) = 1
x1
u(x) where x1 =

√
t21 + t22 + t23.

Using (2.1) and polar coordinates we have∫
0<|ŷ|<R

V (|ŷ|)|∇v|2dx̂

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)
( 1

x1

)2[|∇RNu|2 +
|u|2

x2
1

− 2
1

x1
u
∂u

∂x1

]
x2

1 dx1 dw dz

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)
[
|∇RNu|2 +

|u|2

x2
1

− 1

x1

∂|u|2

∂x1

]
dx1 dw dz.

By integrations by parts, we obtain

−
∫ √R2−w|2

0

V (|y|) 1

x1

∂|u|2

∂x1
dx1

=

∫ √R2−w|2

0

V (|y|)|u|2
∂ 1
x1

∂x1
dx1 +

∫ √R2−w|2

0

|u|2

x1

∂V (|y|)
∂x1

dx1

= −
∫ √R2−w|2

0

V (|y|) |u|
2

x2
1

dx1 +

∫ √R2−w|2

0

|u|2

x1
V ′(|y|)x1

|y|
dx1.

Hence∫
0<|ŷ|<R

V (|ŷ|)|∇v|2dx̂ = |S2|
∫
{0<|y|<R}∩RN+

V (|y|)
[
|∇u|2 +

V ′(|y|)
|y|

|u|2
]
dx. (3.1)

On the other hand, using polar coordinate agains, we have∫
0<|ŷ|<R

W (|ŷ|)|v|2dx̂ = |S2|
∫
{0<|y|<R}∩RN+

W (|y|)
( 1

x1

)2|u|2x2
1 dx1 dw dz

= |S2|
∫
{0<|y|<R}∩RN+

W (|y|)|u|2 dx
(3.2)



EJDE-2020/75 CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES 7

and ∫
0<|ŷ|<R

V (|ŷ|)
∣∣∣∇( v

ϕ(|ŷ|)

)∣∣∣2ϕ2(|ŷ|)dx̂

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣∇( u

ϕ(|y|)
1

x1

)∣∣∣2ϕ2(|y|)x2
1 dx.

(3.3)

Note that by applying Theorem 1.3 to the Bessel pair (rk+1V, rk+1W ), we obtain∫
0<|ŷ|<R

V (|ŷ|)|∇v|2dx̂ =

∫
0<|ŷ|<R

W (|ŷ|)|v|2dx̂

+

∫
0<|ŷ|<R

V (|ŷ|)
∣∣∣∇( v

ϕ(|ŷ|)

)∣∣∣2ϕ2(|ŷ|)dx̂.

Hence, from (3.1), (3.2) and (3.3), we deduce that∫
{0<|y|<R}∩RN+

V (|y|)|∇u|2 dx

−
∫
{0<|y|<R}∩RN+

[
W (|y|)− V ′(|y|)

|y|
]
|u|2 dx

=

∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣∇( u

ϕ(|y|)
1

x1

)∣∣∣2ϕ2(|y|)x2
1 dx.

Next, from (2.2), we obtain∫
0<|ŷ|<R

V (|ŷ|)
∣∣ ŷ
|ŷ|
· ∇ŷv

∣∣2dx̂
= |S2|

∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣ 1

x1

( y
|y|
· ∇yu−

u

|y|

)
|2x2

1 dx

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣( y|y| · ∇yu− u

|y|

)∣∣∣2 dx
= |S2|

∫
{0<|y|<R}∩RN+

V (|y|)| y
|y|
· ∇yu|2 dx+ |S2|

∫
{0<|y|<R}∩RN+

V (|y|) |u|
2

|y|2
dx

− 2|S2|
∫
{0<|y|<R}∩RN+

V (|y|)
( y
|y|
· ∇yu

) u
|y|

dx.

Noting that by polar coordinates and integration by parts, we obtain

− 2

∫
{0<|y|<R}∩RN+

V (|y|)( y
|y|
· ∇yu)

u

|y|
dy

= −2

∫ R

0

∫
Sk−1
+

V (r)urur
k−2 dσ dr

= −
∫ R

0

∫
Sk−1
+

V (r)
d|u|2

dr
rk−2 dσ dr

=

∫ R

0

∫
Sk−1
+

|u|2 d
dr

(V (r)rk−2) dσ dr

=

∫ R

0

∫
Sk−1
+

|u|2
[V ′(r)

r
+ (k − 2)

V (r)

r2

]
rk−1 dσ dr



8 N. T. DUY, H. B. NGUYEN EJDE-2020/75

=

∫
{0<|y|<R}∩RN+

[V ′(|y|)
|y|

+ (k − 2)
V (|y|)
|y|2

]
|u|2 dx.

Hence ∫
0<|ŷ|<R

V (|ŷ|)
∣∣∣ ŷ|ŷ| · ∇ŷv∣∣∣2dx̂

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)| y
|y|
· ∇yu|2 dx

+ |S2|
∫
{0<|y|<R}∩RN+

[V ′(|y|)
|y|

+ (k − 1)
V (|y|)
|y|2

]
|u|2 dx.

(3.4)

Similarly,∫
0<|ŷ|<R

W (|ŷ|)|v|2dx̂ = |S2|
∫
{0<|y|<R}∩RN+

W (|y|)( 1

x1
)2|u|2x2

1 dx1 dw dz

= |S2|
∫
{0<|y|<R}∩RN+

W (|y|)|u|2 dx
(3.5)

and ∫
0<|ŷ|<R

V (|ŷ|)| ŷ
|ŷ|
· ∇ŷ(

v

ϕ(|ŷ|)
)|2ϕ2(|ŷ|)dx̂

= |S2|
∫
{0<|y|<R}∩RN+

V (|y|)| y
|y|
· ∇y(

u

ϕ(|y|)
1

x1
)|2ϕ2(|y|)x2

1 dx.

(3.6)

Applying Theorem 1.3 to the Bessel pair (rk+1V, rk+1W ) we obtain∫
0<|ŷ|<R

V (|ŷ|)| ŷ
|ŷ|
· ∇ŷv|2dx̂−

∫
0<|ŷ|<R

W (|ŷ|)|v|2dx̂

=

∫
0<|ŷ|<R

V (|ŷ|)
∣∣ ŷ
|ŷ|
· ∇ŷ(

v

ϕ(|ŷ|)
)
∣∣2ϕ2(|ŷ|)dx̂.

Then from (3.4), (3.5) and (3.6) we obtain∫
{0<|y|<R}∩RN+

V (|y|)| y
|y|
· ∇yu|2 dx

−
∫
{0<|y|<R}∩RN+

[
W (|y|)− V ′(|y|)

|y|
− (k − 1)

V (|y|)
|y|2

]
|u|2 dx

=

∫
{0<|y|<R}∩RN+

V (|y|)
∣∣∣ y|y| · ∇y( u

ϕ(|y|)
1

x1

)∣∣∣2ϕ2(|y|)x2
1 dx.

�

4. Some consequences of our main result

Now we list a few applications of our results. First, since (rk+1, k
2

4 r
k−1) is a

Bessel pair on (0,∞) with ϕ = r−
k
2 , from Theorem 1.5 we deduce the following

result.

Corollary 4.1. For u ∈ C∞0 (RN+ ) we have∫
RN+
|∇u|2 dx− (

k

2
)2

∫
RN+

|u|2

|y|2
dx =

∫
RN+

1

|y|k
∣∣∣∇(|y|k/2u 1

x1

)∣∣∣2x2
1 dx



EJDE-2020/75 CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES 9

and ∫
RN+

∣∣ y
|y|
· ∇yu

∣∣2 dx− (k − 2

2

)2 ∫
RN+

|u|2

|y|2
dx

=

∫
RN+

1

|y|k
∣∣ y
|y|
· ∇y(|y|k/2u 1

x1
)
∣∣2x2

1 dx.

We note that by Theorem 1.3,∫
RN
|∇u|2 dx−

(k − 2

2

)2 ∫
RN

|u|2

|y|2
dx =

∫
RN

1

|y|k−2
|∇(|y|

k−2
2 u)|2 dx.

Hence, when we restrict the domain to half-spaces, the optimal constant of the
cylindrical Hardy inequality has been improved from (k−2

2 )2 to (k2 )2.

More generally, since (rk+1−α, (k−α)2

4 rk−1−α) is a Bessel pair on (0,∞) with

ϕ = r−
k−α

2 , we obtain the following result.

Corollary 4.2. For u ∈ C∞0 (RN+ ),∫
RN+

|∇u|2

|y|α
dx−

[(k − α
2

)2
+ α

] ∫
RN+

|u|2

|y|2+α
dx =

∫
RN

1

|y|k
|∇(|y|

k−α
2 u

1

x1
)|2x2

1 dx

and ∫
RN+

| y|y| · ∇yu|
2

|y|α
dx−

[(k − α
2

)2
+ α− (k − 1)

] ∫
RN+

|u|2

|y|2+α
dx

=

∫
RN

1

|y|k
∣∣∣ y|y| · ∇y(|y| k−α2 u

1

x1

)∣∣∣2x2
1 dx.

We note again that by Theorem 1.3,∫
RN

|∇yu|2

|y|α
dx−

(k − 2− α
2

)2 ∫
RN

|u|2

|y|2+α
dx =

∫
RN

1

|y|k−2
|∇(|y|

k−2−α
2 u)|2 dx.

Hence, in this case, the sharp constant of the cylindrical Hardy type inequality
has been improved from (k−2−α

2 )2 to (k−α2 )2 + α when we consider the functions

on half-spaces. Now, since (rk+1 1
rk
, rk+1 1

4rk+2| log r
R |2

) is a Bessel pair on (0, R)

with ϕ =
√
| log(r/R)|. By Theorem 1.5, we obtain the cylindrical critical Hardy

inequalities on half-space.

Corollary 4.3. For u ∈ C∞0 ({0 < |y| < R} ∩ RN+ ):∫
{0<|y|<R}∩RN+

|∇u(x)|2

|y|k
dx−

∫
{0<|y|<R}∩RN+

[1

4

1

| log R
|y| |2

+ k
] |u(x)|2

|y|k+2
dx

=

∫
{0<|y|<R}∩RN+

1

|y|k
log

R

|y|

∣∣∣∇( u(x)√
log R
|y|

1

x1

)∣∣∣2x2
1 dx

and∫
{0<|y|<R}∩RN+

| y|y| · ∇yu(x)|2

|y|k
dx−

∫
{0<|y|<R}∩RN+

[1

4

1

| log R
|y| |2

+ 1
] |u(x)|2

|y|k+2
dx

=

∫
{0<|y|<R}∩RN+

1

|y|k
log

R

|y|

∣∣∣ y|y| · ∇y(
u(x)√
log R
|y|

1

x1
)
∣∣∣2x2

1 dx.



10 N. T. DUY, H. B. NGUYEN EJDE-2020/75

These versions of the cylindrical critical Hardy inequalities on half-spaces seem
new in the literature.

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Nguyen Tuan Duy

Department of Fundamental Sciences, University of Finance-Marketing, 2/4 Tran Xuan
Soan St., Tan Thuan Tay Ward, Dist. 7, Ho Chi Minh City, Vietnam

Email address: nguyenduy@ufm.edu.vn



12 N. T. DUY, H. B. NGUYEN EJDE-2020/75

Huy Bac Nguyen (Corresponding Author)

Faculty of Electrical Engineering & Computer Science, Technical University of Os-

trava, Czech Republic
Email address: huy.bac.nguyen.st@vsb.cz


	1. Introduction
	2. Some useful calculations
	3. Proof of main results
	4. Some consequences of our main result
	References