factorizations and hardy’s type identities and ...guozhenlu/papers/lamluzhang-cvp… · on...

Calc. Var. (2019) 58:183https://doi.org/10.1007/s00526-019-1633-x Calculus of Variations

Factorizations and Hardy’s type identities and inequalities onupper half spaces

Nguyen Lam1,2 · Guozhen Lu3 · Lu Zhang4,5

Received: 19 January 2019 / Accepted: 31 August 2019 / Published online: 4 October 2019© Springer-Verlag GmbH Germany, part of Springer Nature 2019

AbstractMotivated and inspired by the improvedHardy inequalities studied in their well-knownworksby Brezis and Vázquez (Rev Mat Univ Complut Madrid 10:443–469, 1997) and Brezis andMarcus (Ann Scuola Norm Sup Pisa Cl Sci 25(1–2):217–237, 1997), we establish in thispaper several identities that imply many sharpened forms of the Hardy type inequalitieson upper half spaces {xN > 0}. We set up these results for the distance to the origin, thedistance to the boundary of any strip R

N−1 × (0, R) and the distance to the hyperplane{xN = 0}, using both the usual full gradient and radial derivative (in the case of distanceto the origin) or only the partial derivative ∂u

∂xN(in the case of distance to the boundary of

the strip or hyperplane). One of the applications of our main results is that when � is thestrip R

N−1 × (0, 2R), the bound λ (�) given by Brezis and Marcus in Brezis and Marcus

(1997) can be improved toz20R2 , where z0 = 2.4048 . . . is the first zero of the Bessel function

J0 (z). Our approach makes use of the notion of Bessel pairs introduced by Ghoussoub andMoradifam (MathAnn349(1):1–57, 2011) and (Functional inequalities: newperspectives andnew applications. Mathematical Surveys and Monographs, American Mathematical Society,Providence, 2013) and the method of factorizations of differential operators. In particular,our identities and inequalities offer sharpened and more precise estimates of the secondremainder term in the existing Hardy type inequalities on upper half spaces in the literature,including the Hardy-Sobolev-Maz’ya type inequalities.

Mathematics Subject Classification 26D10 · 46E35 · 35A23

Communicated by H.Brezis.

N. L. was partially supported by a fellowship from the Pacific Institute for the Mathematical Sciences. G. L.was partially supported by a Grant from the Simons Foundation.

B Guozhen [email protected]

B Lu [email protected]

Extended author information available on the last page of the article

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183 of 31 N. Lam et al.

1 Introduction

The classical Hardy inequalities

∫

�

|∇u|2 dx ≥ Copt (�)

∫

�

|u|2d (x)2

dx , u ∈ C∞0 (�) , (1.1)

where � ⊂ RN , N ≥ 2, d(x) is the distance from x to a certain surface, have attracted

extensive attention because of their important roles in many branches of mathematics. Thereis a vast amount of literature on the Hardy type inequalities together with their variations andgeneralizations. It is impossible to give a complete list on the subject. Hence, we just refer theinterested reader to [32,49] for historical backgrounds and to the monographs [3,33,34,55],that are standard references on the topic.

The principal aim of this note is to use the factorizations of differential operators to studythe Hardy type inequalities on half-spaces. For a comprehensive review, the history andproperties of the factorization method, the interested reader is referred to [24]. We note herethat factorizing differential equations was used in [26] to prove the classical Hardy inequalityand in [25] for the radial and logarithmic refinements of the Hardy inequality. In Gesztesy[24], the authors applied factorizations of singular, even-order partial differential operatorsto provide simple proofs for several Hardy-Rellich type inequalities. Recently in [38,39,42–45], the Hardy operator has been decomposed into or bounded by the product of differentialoperators and thus higher order Hardy-Sobolev-Maz’ya inequalities and their borderline caseof Hardy-Adams inequalities have been established using the Fourier analysis techniqueson hyperbolic spaces, among other techniques (see also the first order Hardy-Trudinger-Moser type inequalities in [41,61,63]). More recently, the factorization method was used toobtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups andweighted Hardy inequalities on general homogeneous groups in [57].

1.1 Hardy inequalities with distance to the origin

When 0 ∈ � and d (x) = |x | is the distance to the origin, the Hardy inequalities (1.1) havebeen investigated extensively and intensively:

∫

�

|∇u|2 dx ≥(N − 2

2

)2 ∫

�

|u|2|x |2 dx , u ∈ C∞

0 (�) . (1.2)

The fact that the optimal constant( N−2

2

)2, N ≥ 3, is not achieved in H1

0 (�) has beenwell understood. On domains having 0 on their boundary, the situation is more complicated.Indeed, in this case, it was proved that the best constant Copt (�) can be anywhere between( N−2

2

)2and

( N2

)2. Moreover, it can be attained by nontrivial functions in H1

0 (�) as long as

Copt (�) <( N2

)2. In particular, on the half-spaces RN+ = {

x ∈ RN : xN > 0

},Copt

(R

N+) =( N

2

)2 : ∫

RN+

|∇u|2 dx ≥(N

2

)2 ∫

RN+

|u|2|x |2 dx , u ∈ C∞

0

(R

N+). (1.3)

See, for instance [14,17,19,53].

123

Hardy type identities and inequalities on half spaces of 31 183

When 0 ∈ �, there has been considerable efforts to improve the Hardy inequalities. One

possible way is to find extra terms to add to the potential( N−2

2

)2 1|x |2 . The first result in

this direction appeared in [12] where in order to investigate the stability of certain singularsolutions of nonlinear elliptic equations, Brezis and Vázquez proved on bounded domainsthat

Theorem A (Brezis-Vázquez [12]). For any bounded domain � ⊂ RN , N ≥ 2, and everyu ∈ H1

0 (�),

∫

�

|∇u|2 dx −(N − 2

2

)2 ∫

�

|u|2|x |2 dx ≥ z20ω

2NN |�|− 2

N

∫

�

|u|2 dx (1.4)

where z0 = 2.4048 . . . is the first zero of the Bessel function J0 (z). The constant z20ω2NN |�|− 2

N

is optimal when � is a ball but is not achieved in H10 (�).

It is also conjectured by Brezis and Vázquez that (see [12]) z20ω2NN |�|− 2

N

∫

�

|u|2 dx is just

the first term of an infinite series of extra terms that can be added to the RHS of (1.4). Theresults of Brezis and Vázquez in [12] have generated a noticeable amount of interest.We referthe reader to, for instance [1,6,7,23,29,52,62]. See also the book [28] for more discussionsand developments in this aspect. We just state here a result by Ghoussoub and Moradifamin [27] that extends and unifies several results in this direction (see also their beautiful book[28]):

Theorem B (Ghoussoub-Moradifam [27]). Assume N ≥ 1. Let 0 < R ≤ ∞, V and W be

positive C1−functions on (0, R) such that

R∫

0

1r N−1V (r)

dr = ∞ and

R∫

0

r N−1V (r) dr < ∞.

If(r N−1V , r N−1W

)is a Bessel pair on (0, R), then for all u ∈ C∞

0 (BR)

∫

BR

V (|x |) |∇u|2 dx ≥∫

BR

W (|x |) |u|2 dx. (1.5)

Also, if (1.5) holds for all u ∈ C∞0 (BR), then

(r N−1V , r N−1cW

)is a Bessel pair on (0, R)

for some c > 0.

Here a couple ofC1−functions (V ,W ) is aBessel pair on (0, R) if the ordinary differentialequation (

V y′)′ + Wy = 0

has a positive solution on the interval (0, R).Another way to improve the Hardy type inequalities is to replace the usual∇ by x

|x | ·∇. We

note that x|x | ·∇ is just the radial derivative. Indeed, in the polar coordinate (r , σ ) =

(|x | , x

|x |),

we have x|x | · ∇u (x) = ∂r u (rσ) =: Ru. We also mention here that the operator R has

appeared naturally in the literature. Indeed, much research has been devoted to investigatingthe functional and geometric inequalities on general homogeneous groups. However, sincethese spaces do not have to be stratified or even graded, the concept of horizontal gradientsdoes not make sense. Thus, one may want to work with the full gradient. On the other

123


hand, unless the homogeneous groups are Abelian, the full gradient is not homogeneous.Nevertheless, on the homogeneous groups, the operatorR is homogeneous of order −1 andthus, is reasonable to work on. Actually, the Hardy type inequalities with radial derivativehave been studied intensively recently. See [31,35,36,58], to name just a few. For instance,in [9,47], the following equalities have been set up to provide a direct understanding to thevalidity as well as the nonexistence of optimizers for the Hardy inequalities:

∫

RN

∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx =(N − 2

2

)2 ∫

RN

|u|2|x |2 dx +

∫

RN

∣∣∣∣ x

|x | · ∇u + N − 2

2

u

|x |∣∣∣∣2

dx

∫

RN

|∇u|2 dx =(N − 2

2

)2 ∫

RN

|u|2|x |2 dx +

∫

RN

∣∣∣∣∇u + N − 2

2

u

|x |x

|x |∣∣∣∣2

dx .

A version of Theorem B with radial derivative was also set up recently in [16].For the improvements of the Hardy type inequalities with boundary singularities, the

interested reader is referred to [6–8,14,17,20,22,40,43,46,51,56,59,60], to name just a few.We note here that a sharp version of the Hardy inequality for fractional Sobolev spaces onhalf-spaces was investigated in [22]. In [6,7], Beckner established Hardy type inequalitieson the entire spaces and half-spaces using Fourier analysis and a nonlinear generalization ofthe Stein-Weiss lemma for non-unimodular groups.

Our first primary goal of this article is to employ the method of factorization to investigatea version of Theorem B in the setting of half-spaces. More precisely, we will show that

Theorem 1.1 Let 0 < R ≤ ∞, V and W be positive C1−functions on (0, R). If(r N+1V , r N+1W

)is a Bessel pair on (0, R), then for u ∈ C∞

0

(B(N )R ∩ R

N+)

:∫

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

ϕ

1

xN

)∣∣∣∣2

ϕ2x2Ndx

and∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −∫

B(N )R ∩RN+

[W (|x |) − V ′ (|x |)

|x | − (N − 1)

|x |2 V (|x |)]

|u|2 dx

=∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇(1

ϕ

u

xN

)∣∣∣∣2

ϕ2x2Ndx .

Here ϕ = ϕr N+1V ,r N+1W ;R is the positive solution of

(r N+1V (r) y′(r)

)′ + r N+1W (r) y(r) = 0

on the interval (0, R).

Asa consequence ofTheorem1.1,weobtain the following type inequalities on half-spaces.

123


Corollary 1.1 If(r N+1V , r N+1W

)is aBessel pair on (0, R), then for u ∈ C∞

0

(B(N )R ∩ R

N+):

∫

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 |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx ≥∫

B(N )R ∩RN+

[W (|x |) − V ′ (|x |)

|x | − (N − 1)

|x |2 V (|x |)]

|u|2 dx

with the ground state xNϕ (x).

By applyingTheorem1.1 to theBessel pair(r N+1,

( N2

)2r N−1

)withϕ = r− N

2 on (0,∞),

we deduce the following identities that provide a straightforward interpretation of (1.3):

Corollary 1.2 For u ∈ C∞0

(R

N+), there holds

∫

RN+

|∇u|2 dx −(N

2

)2 ∫

RN+

|u|2|x |2 dx =

∫

RN+

∣∣∣∣∇(

|x | N2 u

xN

)∣∣∣∣2

|x |−N x2Ndx (1.6)

∫

RN+

∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − 2

2

)2 ∫

RN+

|u|2|x |2 dx =

∫

RN+

∣∣∣∣ x

|x | · ∇(

|x | N2 u

xN

)∣∣∣∣2

|x |−N x2Ndx .

(1.7)

We note that xN |x |− N2 is not in L2

(R

N+). Indeed,

∫

RN+

x2N |x |−N dx =∞∫

0

∫

RN−1

x2N∣∣x2N + |y|2∣∣− N

2 dydxN

=∣∣∣SN−2

∣∣∣∞∫

0

∞∫

0

x2N∣∣x2N + r2

∣∣− N2 r N−2drdxN .

By setting r = xN t , we get

∞∫

0

∞∫

0

x2N∣∣x2N + r2

∣∣− N2 r N−2drdxN =

∞∫

0

∞∫

0

∣∣1 + t2∣∣− N

2 t N−2xNdtdxN

=∞∫

0

xNdxN

∞∫

0

∣∣1 + t2∣∣− N

2 t N−2dt

which is divergent. Hence, the best constants( N2

)2in (1.6) and

( N−22

)2in (1.7) cannot be

attained by nontrivial functions in H10

(R

N+). Nevertheless, from the equality (1.6), we deduce

that cxN |x |− N2 , c ∈ C, can play the role of the “virtual” ground state in the sense of Frank

and Seiringer [21].

Remark 1.1 It is also worthy to note that the best constant for the Hardy inequality on half-spaces with radial derivative is exactly the same as the optimal constant of the one on the

123


whole space. This is in contrast to the case of full gradient where the sharp constant of the

Hardy inequality on half-spaces with usual gradient is bumped to( N2

)2on R

N+ .

Using Theorem 1.1, we can derive many new and interesting Hardy type inequalities onhalf-spaces. Indeed, we can deduce as many the Hardy inequalities on R

N+ as we can formthe Bessel pairs. The book [28] provides various examples and properties about Bessel pairs.In the next section, we will state a few typical versions of the Hardy type inequalities onhalf-spaces that seem new in the literature and have potential applications in the study of thestability of certain singular solutions of nonlinear elliptic equations on half-spaces.

We illustrate below some particular applications of our Theorem 1.1 which provide thefollowing sharpened inequalities on half balls or half spaces in the spirit ofBrezis andVázquez[12].

Theorem 1.2 For any 0 < R ≤ ∞, we have for u ∈ C∞0

(B(N )R ∩ R

N+)

:∫

B(N )R ∩RN+

|∇u|2 dx −(N

2

)2 ∫

B(N )R ∩RN+

|u|2|x |2 dx

= z20R2

∫

B(N )R ∩RN+

|u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣∇(

|x | N2J0;R(|x |)

u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N2

xN

∣∣∣∣∣2

dx

≥ z20R2

∫

B(N )R ∩RN+

|u|2 dx,

and∫

B(N )R ∩RN+

∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − 2

2

)2 ∫

B(N )R ∩RN+

|u|2|x |2 dx

= z20R2

∫

B(N )R ∩RN+

|u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣x

|x | · ∇(

|x | N2J0;R(|x |)

u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N2

xN

∣∣∣∣∣2

dx

≥ z20R2

∫

B(N )R ∩RN+

|u|2 dx .

We can also obtain the following versions of the celebrated Heisenberg’s uncertaintyprinciple on half-spaces as applications of our results:

∫

RN+

|u|2 dx

≤

⎛⎜⎜⎝

∫

RN+

|u|2|x |2 dx

⎞⎟⎟⎠

12⎛⎜⎜⎝

∫

RN+

|x |2 |u|2 dx

⎞⎟⎟⎠

12

123


≤ 2

N − 2

⎛⎜⎜⎝

∫

RN+

∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx

⎞⎟⎟⎠

12⎛⎜⎜⎝

∫

RN+

|x |2 |u|2 dx

⎞⎟⎟⎠

12

and ∫

RN+

|u|2 dx

≤

⎛⎜⎜⎝

∫

RN+

|u|2|x |2 dx

⎞⎟⎟⎠

12⎛⎜⎜⎝

∫

RN+

|x |2 |u|2 dx

⎞⎟⎟⎠

12

≤ 2

N

⎛⎜⎜⎝

∫

RN+

|∇u|2 dx

⎞⎟⎟⎠

12⎛⎜⎜⎝

∫

RN+

|x |2 |u|2 dx

⎞⎟⎟⎠

12

.

1.2 Hardy inequalities with distance to a surface of codimension 1

Another type of the Hardy inequalities that has also received much attention in the literatureis the case when d (x) is the distance to a surface K of codimention 1. The most notableexample is d (x) = d (x, ∂�) is the distance to the boundary. Here, � is a domain in R

N

with Lipschitz boundary. In this case, the optimal constant

Copt (�) := infu∈H1

0 (�)

∫

�

|∇u|2 dx∫

�

|u|2d(x,∂�)2

dx

depends on the domain. In general, Copt (�) ≤ 14 when ∂� has a tangent plane at least at

one point. There exists smooth bounded domains such that Copt (�) < 14 , but for convex

domains Copt (�) = 14 . It is interesting to mention that the optimal constant Copt (�) can be

attained if and only if Copt (�) < 14 . See, for instance [4,10,11,15,48]. Actually, Brezis and

Marcus proved in [10] the following result to explain explicitly this situation on boundedconvex domain:

Theorem C (Brezis-Marcus [10]). For every smooth domain of class C2, there exists λ (�) ∈R such that∫

�

|∇u|2 dx − 1

4

∫

�

|u|2d (x, ∂�)2

dx ≥ λ (�)

∫

�

|u|2 dx, u ∈ C∞0 (�) . (1.8)

Moreover, for convex domains,

λ (�) ≥ 1

4diam2 (�).

123


They also questioned in [10] whether the diameter could be replaced by the volume of

�. An affirmative answer was posed in [30] where it was showed that λ (�) ≥ NN−2N ω

2NN−1

4|�| 2N

.

In [18], the authors improved the bound in [10] to λ (�) ≥ 3Dint (�)2

where Dint (�) :=2 supx∈� d (x, ∂�) is the interior diameter. In [2], Avkhadiev and Wirths established a sharp

bound λ (�) ≥ λ20

(supx∈� d(x,∂�))2 , λ0 = 0.940 . . . is the first zero in (0,∞) of the function

J0 (x) + 2x J ′0 (x), using one-dimensional inequalities. We refer the interested reader to

the monograph [3] for further developments in this direction. We brieftly note here that

as a consequence of our main results, the constant λ (�) can be improved toz20R2 when

� = RN−1 × (0, 2R).

The next goal of this paper is to investigate the general versions of the Hardy inequalitywith distance to the boundary on R

N+ . We will use the method of factorizing differentialoperators and the notion of Bessel pairs to consolidate and improve many existing resultsin the literature. More precisely, let 0 < R ≤ ∞ and let A(N )

R = RN−1 × (0, R) be a strip

in RN . Denote dR (x) = min {xN , 2R − xN } (with the convention that d∞ (x) = xN ). Note

that dR (x) is the distance from x to the boundary of the strip A(N )2R . Then we will prove that

Theorem 1.3 Let0 < R ≤ ∞. If (V ,W ) is aBessel pair on (0, R), then for u ∈ C∞0

(A(N )2R

):

∫

A(N )2R

V (dR (x)) |∇u|2 dx −∫

A(N )2R

W (dR (x)) |u|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣∇(

u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx,

and ∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx −∫

A(N )2R

W (dR (x)) |u|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂

∂xN

(u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx .

We note that for any 0 < R < ∞,

(1, 1

41r2

+ z20R2

)is a Bessel pair on (0, R) with

ϕ = √r J0

( r z0R

) = √r J0;R (r). Hence we get the following Hardy inequalities in the spirit

of Brezis-Marcus:

Theorem 1.4 For any 0 < R < ∞, we have for u ∈ C∞0

(A(N )2R

):

∫

A(N )2R

|∇u|2 dx − 1

4

∫

B(N )R ∩RN+

|u|2dR (x)2

dx

= z20R2

∫

A(N )2R

|u|2 dx +∫

A(N )2R

∣∣∣∣∇(

u√dR (x)J0;R(dR (x))

)∣∣∣∣2 ∣∣J0;R(dR (x))

∣∣2 dR (x) dx

123


and∫

A(N )2R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx − 1

4

∫

B(N )R ∩RN+

|u|2dR (x)2

dx

= z20R2

∫

A(N )2R

|u|2 dx +∫

A(N )2R

∣∣∣∣ ∂

∂xN

(u√

dR (x)J0;R(dR (x))

)∣∣∣∣2 ∣∣J0;R(dR (x))

∣∣2 dR (x) dx .

Hence, when the domain is the strip A(N )2R , we improve the constant λ (�) in Theorem C

toz20R2 . This theorem can be considered as a special case of Corollary 3.1 in Sect. 3.

Another interesting application of our Theorem 1.3 is as follows: Since(rα+1, α2

4 rα−1

)

is a Bessel pair on (0,∞) with ϕ = r− α2 , we obtain the following identities:

Corollary 1.3 For u ∈ C∞0

(R

N+), there holds

∫

RN+

|∇u|2 xα+1N dx − α2

4

∫

RN+

|u|2 xα−1N dx =

∫

RN+

∣∣∣∇(x

α2N u

)∣∣∣2 xNdx

∫

RN+

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

xα+1N dx − α2

4

∫

RN+

|u|2 xα−1N dx =

∫

RN+

∣∣∣∣∣∣∣∂

(x

α2N u

)

∂xN

∣∣∣∣∣∣∣

2

xNdx .

By choosing α = −1, we get∫

RN+

|∇u|2 dx − 1

4

∫

RN+

|u|2x2N

dx ≥ 0.

Actually, we also have the following Hardy-Sobolev-Maz’ya inequality: (see, for instance[50]):

∫

RN+

|∇u|2 dx − 1

4

∫

RN+

|u|2x2N

dx ≥ 1

C2 (N , 1)

⎛⎜⎜⎝

∫

RN+

|u| 2(N+1)N−1 x

− 2N−1

N dx

⎞⎟⎟⎠

N−1N+1

, u ∈ C∞0

(R

N+).

(1.9)Here C (N , 1) is sharp and is given in (1.10). We will show that by combining the identitiesin Corollary 1.3 and the optimal Sobolev inequality with monomial weights, we can derivea sharp version of the Hardy-Sobolev-Maz’ya type inequalities.

Theorem 1.5 Let α ∈ R. Then we have∫

RN+

|∇u|2 xα+1N dx − α2

4

∫

RN+

|u|2 xα−1N dx

≥ 1

C2 (N , 1)

⎛⎜⎜⎝

∫

RN+

|u| 2(N+1)N−1 x

N−1+α(N+1)N−1

N dx

⎞⎟⎟⎠

N−1N+1

.

123


The optimal constant C (N , 1) given in (1.10) is optimal and can be achieved when u (x) =x−α/2N

(a + b |x |2)(1−N )/2

, a and b are arbitrary positive constants.

Indeed, the optimal Sobolev inequality with monomial weights in [13,54] asserts that

⎛⎜⎜⎝

∫

RN+

|∇v|2 x ANdx

⎞⎟⎟⎠

12

≥ 1

C (N , A)

⎛⎜⎜⎝

∫

RN+

|v|2∗A x A

Ndx

⎞⎟⎟⎠

12∗A

where 2∗A = 2(N+A)

N+A−2 and the best constant C (N , A) is achieved by Ua,b (x) =(a + b |x |2)1− N+A

2 , a and b are arbitrary positive constants:

C (N , A) = (N + A)

(

( A+12

)2

(1 + N+A

2

)) 1

N+A

× (N + A)1

N+A − 32

(1

N + A − 2

) 12(

2 (N + A)

( N+A

2

)

( N+A2

)) 1

N+A

. (1.10)

Using this result, we get for any α ∈ R that

∫

RN+

|∇u|2 xα+1N dx − α2

4

∫

RN+

|u|2 xα−1N dx

=∫

RN+

∣∣∣∇(x

α2N u

)∣∣∣2 xNdx

≥ 1

C2 (N , 1)

⎛⎜⎜⎝

∫

RN+

∣∣∣x α2N u

∣∣∣2(N+1)N−1

xNdx

⎞⎟⎟⎠

N−1N+1

= 1

C2 (N , 1)

⎛⎜⎜⎝

∫

RN+

|u| 2(N+1)N−1 x

1+ α(N+1)N−1

N dx

⎞⎟⎟⎠

N−1N+1

.

Remark 1.2 We remark here that the method in [50, Section 2.7] is dimension reduction.Hence the argument there doesn’t work for arbitrary real number α.

Our final goal of this paper is to set up Hardy type inequalities with distance to thehyperplane {xN = 0} on R

N+ .

Theorem 1.6 If (V ,W ) is a Bessel pair on (0, R), then for u ∈ C∞0

(A(N )R

):

123


∫

A(N )R

V (xN ) |∇u|2 dx −∫

A(N )R

W (xN ) |u|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣∇(

u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx,

and ∫

A(N )R

V (xN )

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

−∫

A(N )R

W (xN ) |u|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣ ∂

∂xN

(u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx .

For instance, we can get the following result in the spirit of Brezis-Vázquez and Brezis-Marcus:

Corollary 1.4 For any 0 < R < ∞, we have for u ∈ C∞0

(A(N )R

):

∫

A(N )R

|∇u|2 dx − 1

4

∫

A(N )R

|u|2x2N

dx

= z20R2

∫

A(N )R

|u|2 dx +∫

A(N )R

∣∣∣∣∇(

u√xN J0;R(xN )

)∣∣∣∣2 ∣∣J0;R(xN )

∣∣2 xNdx

and ∫

A(N )R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx − 1

4

∫

A(N )R

|u|2x2N

dx

= z20R2

∫

A(N )R

|u|2 dx +∫

A(N )R

∣∣∣∣∣∣∂

(u√

xN J0;R(xN )

)

∂xN

∣∣∣∣∣∣

2 ∣∣J0;R(xN )∣∣2 xNdx .

Basically, the above results assert that on A(N )R , if we replace the distance to the boundary

by the distance to the hyperplane {xN = 0}, then the constant λ (�) in Theorem C can be

improved toz20R2 .

Furthermore, as another application of Theorem 1.6, we will be able to establish Hardytype identities and inequalities with as many remainder terms as we wish (See Corollary 4.2)in Sect. 4.

We end up this introduction with the following remarks. In our recent work [37], weestablish general geometric Hardy’s inequalities on domains in R

N in the spirit of theirworks by Brezis-Vázquez [12] and Brezis-Marcus [10]. More precisely, we use the notionof Bessel pairs introduced by Ghoussoub and Moradifam [27,28] to establish several Hardyidentities and inequalities with the general distance functions d(x) from x to a surface of anycodimension.

123


Theorem D Let 0 < R ≤ ∞, V and W be positive C1−functions on (0, R). Assumethat |∇d(x)| = 1 and for some α ∈ R, d (x) − α−1

d(x) exists on {0 < d (x) < R} in

the sense of distributions and(rα−1V , rα−1W

)is a Bessel pair on (0, R). Then for

u ∈ C∞0 ({0 < d (x) < R}) :∫

0<d(x)<R

V (d (x)) |∇u (x)|2 dx −∫

0<d(x)<R

W (d (x)) |u (x)|2 dx

=∫

0<d(x)<R

V (d (x)) ϕ2 (d (x))

∣∣∣∣∇(

u (x)

ϕ (d (x))

)∣∣∣∣2

dx

−∫

0<d(x)<R

V (d (x)) |u (x)|2[d (x) − α − 1

d (x)

]ϕ′ (d (x))

ϕ (d (x))dx

and ∫

0<d(x)<R

V (d (x)) |∇d (x) · ∇u (x)|2 dx −∫

0<d(x)<R

W (d (x)) |u (x)|2 dx

=∫

0<d(x)<R

V (d (x)) ϕ2 (d (x))

∣∣∣∣∇d (x) · ∇(

u (x)

ϕ (d (x))

)∣∣∣∣2

dx

−∫

0<d(x)<R

V (d (x)) |u (x)|2[d (x) − α − 1

d (x)

]ϕ′ (d (x))

ϕ (d (x))dx .

Here ϕ is the positive solution of(rα−1V (r) y′(r)

)′ + rα−1W (r) y(r) = 0

on the interval (0, R).

It is worth noting that our distance functions can be understood as the distance to thesurfaces of codimention α ∈ R, and include the distance to the origin (α = N ), the distanceto the boundary (α = 1), and even the distance to surfaces of codimension k ∈ N with1 ≤ k ≤ N , as special cases. As applications of our main results, we are able to obtainthe improved Hardy inequalities in the sense of Brezis-Vázquez [12] and Brezis-Marcus[10] for general distance functions. We also prove a version of the Hardy-Sobolev-Maz’yainequality on (R+)N for the distance functionmin {x1, . . . , xN }. Moreover, the critical Hardyinequalities with general distance functions are also considered. Our results also extendand sharpen earlier results on Hardy’s inequalities with distance functions to surfaces ofcodimension 1 ≤ k ≤ N in the literature by adding nonnegative remainder terms.

The organization of the paper is as follows. In Sect. 2, wewill establish theHardy identitiesand inequalities with distance to the origin on half-spaces and prove Theorem 1.1. We willalso apply this theorem to derive several Corollaries of Hardy type identities and inequalitieson half spaces in the spirit of Brezis and Vázquez. In Sect. 3, we will prove the Hardy typeidentities and inequalities with distance to the boundary of the strips A(N )

2R on half-spaces,namely Theorem 1.3.Wewill also give some applications of Theorem 1.3. As a consequence,we will improve the bounds λ(�) in TheoremC of Brezis andMarcus when� is a strip A(N )

2R .We will also get an extension and improvement of the Hardy-Sobolev-Maz’ya inequality toarbitrary α ≤ 0. Finally, in Sect. 4, we will prove the Hardy type identities and inequalities

123


with distance to the hyperplane {xN = 0}, namely Theorem 1.6. We will also give someapplications of Theorem 1.6.

2 Hardy identities and inequalities with distance to the origin onhalf-spaces

In this section, we will establish the Hardy identities and inequalities with distance to theorigin on half-spaces and prove Theorem 1.1.

2.1 Proof of Theorem 1.1

Proof of Theorem 1.1 Let T = √V∇ − √

V ∇ϕϕ

− √V eN

xN. Then its formal adjoint is T+ =

− div(√V ·) − √

V ∇ϕϕ

· −√V eN

xN·.

Also,

T+T = − div (V∇) + div

(V

∇ϕ

ϕ

)+ V

∇ϕ

ϕ· ∇ + div

(VeNxN

)+ V

eNxN

· ∇

− V∇ϕ

ϕ· ∇ + V

∇ϕ

ϕ· ∇ϕ

ϕ+ V

∇ϕ

ϕ· eNxN

− VeNxN

· ∇ + V∇ϕ

ϕ· eNxN

+ V1

x2N

= − div (V∇) + div

(V

∇ϕ

ϕ

)+ V ′

|x |

+ V

(ϕ′

ϕ

)2

+ Vϕ′

ϕ

1

|x |+ V

ϕ′

ϕ

1

|x |Hence ∫

B(N )R ∩RN+

u (x)(T+Tu

)(x) dx

= −∫

B(N )R ∩RN+

u (x) div (V∇u (x)) dx +∫

B(N )R ∩RN+

div

(V

∇ϕ

ϕ

)|u (x)|2 dx

+∫

B(N )R ∩RN+

V ′

|x | |u (x)|2 dx

+∫

B(N )R ∩RN+

V

(ϕ′

ϕ

)2

|u (x)|2 dx + 2∫

B(N )R ∩RN+

Vϕ′

ϕ

1

|x | |u (x)|2 dx

=∫

B(N )R ∩RN+

V (|x |) |∇u|2 dx +∫

B(N )R ∩RN+

Vϕ′′ + V N+1|x | ϕ′ + V ′ϕ′

ϕ|u (x)|2 dx

123


+∫

B(N )R ∩RN+

V ′

|x | |u (x)|2 dx

=∫

B(N )R ∩RN+

V (|x |) |∇u|2 dx −∫

B(N )R ∩RN+

[W (|x |) − V ′ (|x |)

|x |]

|u|2 dx .

On the other hand, ∫

B(N )R ∩RN+

u (x)(T+Tu

)(x) dx

=∫

B(N )R ∩RN+

|Tu|2 dx

=∫

B(N )R ∩RN+

V (|x |)∣∣∣∣∇

(u

ϕ

1

xN

)∣∣∣∣2

ϕ2x2Ndx .

Hence ∫

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

ϕ

1

xN

)∣∣∣∣2

ϕ2x2Ndx .

Next, we set S = √V x

|x | · ∇ − √V ϕ′

ϕ− √

V 1|x | . Then its formal adjoint is

S+ = −√V

x

|x | · ∇ − √VN − 1

|x | − V ′

2√V

− √V

ϕ′

ϕ− √

V1

|x |= −√

Vx

|x | · ∇ − √V

N

|x | − V ′

2√V

− √V

ϕ′

ϕ

Then we have∫

B(N )R ∩RN+

u (x)(S+Su

)(x) dx

= −∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

Vx

|x | · ∇u

)dx +

∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

Vϕ′

ϕu

)dx

+∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

V1

|x |u)dx

−∫

B(N )R ∩RN+

u

(√V

N

|x | + V ′

2√V

+ √V

ϕ′

ϕ

) √V

x

|x | · ∇udx

123


+∫

B(N )R ∩RN+

(√V

N

|x | + V ′

2√V

+ √V

ϕ′

ϕ

) √V

ϕ′

ϕ|u|2 dx

+∫

B(N )R ∩RN+

(√V

N

|x | + V ′

2√V

+ √V

ϕ′

ϕ

) √V

1

|x | |u|2 dx .

Using the polar coordinate (r , σ ) =(|x | , x

|x |)and the fact that x

|x | · ∇ = ∂r , we get by a

direct computation that

−∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

Vx

|x | · ∇u

)dx

=∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx +∫

B(N )R ∩RN+

u

(x

|x | · ∇u

)V ′

2dx

+∫

B(N )R ∩RN+

u

(x

|x | · ∇u

)VN − 1

|x | dx,

and ∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

Vϕ′

ϕu

)dx

=∫

B(N )R ∩RN+

u

(x

|x | · ∇u

)V

ϕ′

ϕdx +

∫

B(N )R ∩RN+

V

(ϕ′′ϕ − (

ϕ′)2ϕ2

)|u|2 dx

+∫

B(N )R ∩RN+

ϕ′

ϕ

V ′

2|u|2 dx .

and ∫

B(N )R ∩RN+

u√V

x

|x | · ∇(√

V1

|x |u)dx

=∫

B(N )R ∩RN+

(V ′

2 |x | − V

|x |2)

|u|2 dx +∫

B(N )R ∩RN+

u

(x

|x | · ∇u

)V

|x |dx

Hence ∫

B(N )R ∩RN+

u (x)(S+Su

)(x) dx

=∫

B(N )R ∩RN+

[Vϕ′′ + V N+1

|x | ϕ′ + V ′ϕ′

ϕ+ V ′

|x | + VN − 1

|x |2]

|u|2 dx

123


+∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx

=∫

B(N )R ∩RN+

[−W (|x |) + V ′ (|x |)

|x | + (N − 1)

|x |2 V (|x |)]

|u|2 dx

+∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx .

On the other hand,∫

B(N )R ∩RN+

u (x)(S+Su

)(x) dx

=∫

B(N )R ∩RN+

|Su|2 dx

=∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇(1

ϕ

u

xN

)∣∣∣∣2

ϕ2x2Ndx .

Hence, we deduce

∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −∫

B(N )R ∩RN+

[W (|x |) − V ′ (|x |)

|x | − (N − 1)

|x |2 V (|x |)]

|u|2 dx

=∫

B(N )R ∩RN+

V (|x |)∣∣∣∣ x

|x | · ∇(1

ϕ

u

xN

)∣∣∣∣2

ϕ2x2Ndx .

�

2.2 Applications of Theorem 1.1

The pair

(r N+1, r N+1

(( N2

)2r−2 + z20

R2

)), z0 is the first zero of the Bessel function J0, is

a Bessel pair on (0, R) with ϕ = r− N2 J0

( r z0R

) = r− N2 J0;R (r). Hence, as a consequence of

our Theorem 1.1, we get the following version of the Hardy inequality on RN+ in the spirit ofBrezis and Vázquez [12]:

Corollary 2.1 For any R > 0, we have for u ∈ C∞0

(B(N )R ∩ R

N+)

:∫

B(N )R ∩RN+

|∇u|2 dx −(N

2

)2 ∫

B(N )R ∩RN+

|u|2|x |2 dx

123


= z20R2

∫

B(N )R ∩RN+

|u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣∇(

|x | N2J0;R(|x |)

u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N2

xN

∣∣∣∣∣2

dx

≥ z20R2

∫

B(N )R ∩RN+

|u|2 dx, (2.1)

and

∫

B(N )R ∩RN+

∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − 2

2

)2 ∫

B(N )R ∩RN+

|u|2|x |2 dx

= z20R2

∫

B(N )R ∩RN+

|u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣x

|x | · ∇(

|x | N2J0;R(|x |)

u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N2

xN

∣∣∣∣∣2

dx .

A version of (2.1) on the bounded domain containing 0 was proved in [12] and was usedto study the stability of certain singular solutions of nonlinear elliptic equations. Hence, ourresults in the above corollary can be used to investigate the stability of singular solutions ofcertain nonlinear elliptic equations on half-spaces.

Assume 0 ≤ λ ≤ N . Then(r N+1−λ,

( N−λ2

)2r N−1−λ

)is a Bessel pair on (0,∞) with

ϕ = r− N−λ2 . Hence, we get from Theorem 1.1 that

Corollary 2.2 Assume 0 ≤ λ ≤ N. We have for u ∈ C∞0

(R

N+) :

∫

RN+

1

|x |λ |∇u|2 dx −[(

N − λ

2

)2

+ λ

] ∫

RN+

1

|x |λ+2|u|2 dx

=∫

RN+

∣∣∣∣∣∇(

|x | N−λ2 u

xN

)∣∣∣∣∣2

x2N|x |N dx

and

∫

RN+

1

|x |λ∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − λ − 2

2

)2 ∫

RN+

1

|x |λ+2|u|2 dx

=∫

RN+

∣∣∣∣∣x

|x | · ∇(

|x | N−λ2 u

xN

)∣∣∣∣∣2

x2N|x |N dx .

Assume 0 ≤ λ ≤ N . Then

(r N+1−λ,

( N−λ2

)2r N−1−λ + z20

R2 rN+1−λ

)is a Bessel pair on

(0, R) with ϕ = r− N−λ2 J0

( r z0R

) = r− N−λ2 J0;R (r). Hence by Theorem 1.1, we obtain

123


Corollary 2.3 Assume 0 ≤ λ ≤ N. We have for u ∈ C∞0

(B(N )R ∩ R

N+)

:∫

B(N )R ∩RN+

1

|x |λ |∇u|2 dx −[(

N − λ

2

)2

+ λ

] ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= z20R2

∫

B(N )R ∩RN+

1

|x |λ |u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣∇(

|x | N−λ2

J0;R(|x |)u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N−λ

2

xN

∣∣∣∣∣2

dx

≥ z20R2

∫

B(N )R ∩RN+

1

|x |λ |u|2 dx

and ∫

B(N )R ∩RN+

1

|x |λ∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − 2 − λ

2

)2 ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= z20R2

∫

B(N )R ∩RN+

1

|x |λ |u|2 dx +∫

B(N )R ∩RN+

∣∣∣∣∣x

|x | · ∇(

|x | N−λ2

J0;R(|x |)u

xN

)∣∣∣∣∣2 ∣∣∣∣∣

J0;R (|x |)|x | N−λ

2

xN

∣∣∣∣∣2

dx

≥ z20R2

∫

B(N )R ∩RN+

1

|x |λ |u|2 dx .

For any R > 0, 0 ≤ λ ≤ N , and k ∈ N, the pair(r N+1−λ,

( N−λ2

)2r N−1−λ +

r N+1−λPk,R (r))is a Bessel pair on (0, R) with ϕ = r− N−λ

2[X1

( rR

) · · · Xk( rR

)]− 12 where

Pk,R (r) = 1

4

1

r2

k∑j=1

X21

( r

R

)· · · X2

j

( r

R

)

X1 (r) = 1

1 − ln r, Xi (r) = X1 (Xi−1 (r)) .

See [28, Proposition 1.1.1 and Proposition 1.3.1]. Hence, we deduce

Corollary 2.4 For any R > 0, 0 ≤ λ ≤ N, and k ∈ N, we have

∫

B(N )R ∩RN+

1

|x |λ |∇u|2 dx −[(

N − λ

2

)2

+ λ

] ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= 1

4

∫

B(N )R ∩RN+

|u|2|x |λ+2

k∑j=1

X21

( |x |R

)· · · X2

j

( |x |R

)dx

+∫

B(N )R ∩RN+

∣∣∣∣∣∇(

|x | N−λ2

[X1

( |x |R

)· · · Xk

( |x |R

)] 12 u

xN

)∣∣∣∣∣2

123


∣∣∣∣∣∣∣∣xN

|x | N−λ2

[X1

( |x |R

)· · · Xk

( |x |R

)] 12

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

B(N )R ∩RN+

|u|2|x |λ+2

k∑j=1

X21

( |x |R

)· · · X2

j

( |x |R

)dx (2.2)

and ∫

B(N )R ∩RN+

1

|x |λ∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − λ − 2

2

)2 ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= 1

4

∫

B(N )R ∩RN+

|u|2|x |λ+2

k∑j=1

X21

( |x |R

)· · · X2

j

( |x |R

)dx

+∫

B(N )R ∩RN+

∣∣∣∣∣x

|x | · ∇(

|x | N−λ2

[X1

( |x |R

)· · · Xk

( |x |R

)] 12 u

xN

)∣∣∣∣∣2

∣∣∣∣∣∣∣∣xN

|x | N−λ2

[X1

( |x |R

)· · · Xk

( |x |R

)] 12

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

B(N )R ∩RN+

|u|2|x |λ+2

k∑j=1

X21

( |x |R

)· · · X2

j

( |x |R

)dx . (2.3)

The inequality (2.2) was derived in ([59]) in the case λ = 0 using the spherical harmonicdecomposition. Our results here provide the exact remainders and the ground states. The case0 < λ ≤ N is new. (2.3) is completely new in the literature.

Assume 0 ≤ λ ≤ N , R > 0 and ρ > Re.(r N+1−λ,

( N−λ2

)2r N−1−λ +

14r

N−1−λ(log ρ

r

)−2)is a Bessel pair on (0, R) with ϕ = r− N−λ

2(log ρ

r

) 12 . Hence

Corollary 2.5 Assume 0 ≤ λ ≤ N, R > 0 and ρ > Re. We have

∫

B(N )R ∩RN+

1

|x |λ |∇u|2 dx −[(

N − λ

2

)2

+ λ

] ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= 1

4

∫

B(N )R ∩RN+

|u|2

|x |λ+2(log ρ

|x |)2 dx +

∫

B(N )R ∩RN+

∣∣∣∣∣∣∣∣∇

⎛⎜⎜⎝ |x | N−λ

2

(log ρ

|x |) 1

2

u

xN

⎞⎟⎟⎠

∣∣∣∣∣∣∣∣

2 ∣∣∣∣∣∣∣∣

(log ρ

|x |) 1

2

|x | N−λ2

xN

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

B(N )R ∩RN+

|u|2

|x |λ+2(log ρ

|x |)2 dx

123


and∫

B(N )R ∩RN+

1

|x |λ∣∣∣∣ x

|x | · ∇u

∣∣∣∣2

dx −(N − 2 − λ

2

)2 ∫

B(N )R ∩RN+

1

|x |λ+2|u|2 dx

= 1

4

∫

B(N )R ∩RN+

|u|2

|x |λ+2(log ρ

|x |)2 dx

+∫

B(N )R ∩RN+

∣∣∣∣∣∣∣∣x

|x | · ∇

⎛⎜⎜⎝ |x | N−λ

2

(log ρ

|x |) 1

2

u

xN

⎞⎟⎟⎠

∣∣∣∣∣∣∣∣

2 ∣∣∣∣∣∣∣∣

(log ρ

|x |) 1

2

|x | N−λ2

xN

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

B(N )R ∩RN+

|u|2

|x |λ+2(log ρ

|x |)2 dx .

3 Hardy identities and inequalities with distance to the boundary ofdomains on half-spaces

In this section, we will provide the proof for the general Hardy inequality with distance tothe boundary of the strip A(N )

2R . We also derive a general result that implies that the constant

λ (�) in Theorem C can be improved toz20R2 when � is the strip A(N )

2R .


Proof of Theorem 1.3 Let T = √V (dR (x))∇ − √

V (dR (x)) ϕ′(dR(x))ϕ(dR(x)) ∇dR (x). Then its for-

mal adjoint is T+ = − div(√V (dR (x))·) − √

V (dR (x)) ϕ′(dR(x))ϕ(dR(x)) ∇dR (x) ·.

Hence,

T+Tu = − div

(V (dR (x))

[∇u − ϕ′ (dR (x))

ϕ (dR (x))u∇dR (x)

])

− V (dR (x))ϕ′ (dR (x))

ϕ (dR (x))∇dR (x) · ∇u + V (dR (x))

(ϕ′ (dR (x))

ϕ (dR (x))

)2

u

= −V (dR (x)) u − V ′ (dR (x)) ∇dR (x) · ∇u

+ V ′ (dR (x))ϕ′ (dR (x))

ϕ (dR (x))u + V (dR (x))

ϕ′′ (dR (x))

ϕ (dR (x))u.

Thus∫

A(N )2R

u (x)(T+Tu

)(x) dx

=∫

A(N )2R

V (dR (x)) |∇u|2 dx+∫

A(N )2R

V ′ (dR (x)) ϕ′ (dR (x)) + V (dR (x)) ϕ′′ (dR (x))

ϕ (dR (x))|u|2 dx

123


=∫

A(N )2R

V (dR (x)) |∇u|2 dx −∫

A(N )2R

W (dR (x)) |u|2 dx .


A(N )2R

u (x)(T+Tu

)(x) dx

=∫

A(N )2R

|Tu|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣∇(

u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx .

Hence, we deduce∫

A(N )2R

V (dR (x)) |∇u|2 dx −∫

A(N )2R

W (dR (x)) |u|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣∇(

u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx .

Now, let Su = √V (dR (x)) ∂u

∂xN− √

V (dR (x)) ϕ′(dR(x))ϕ(dR(x))

∂dR(x)∂xN

u. Then its formal adjoint

is S+v = − ∂∂xN

(√V (dR (x))v

) − √V (dR (x)) ϕ′(dR(x))

ϕ(dR(x))∂dR(x)∂xN

v.

Therefore,

S+Su

= − ∂

∂xN

(V (dR (x))

∂u

∂xN− V (dR (x))

ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xNu

)

− V (dR (x))ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xN

∂u

∂xN+ V (dR (x))

(ϕ′ (dR (x))

ϕ (dR (x))

)2

u

= −V (dR (x))

(∂2u

∂x2N− ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xN

∂u

∂xN

− ϕ′′ (dR (x)) ϕ (dR (x)) − (ϕ′ (dR (x))

)2ϕ2 (dR (x))

u

)

− V ′ (dR (x))∂dR (x)

∂xN

(∂u

∂xN− ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xNu

)

− V (dR (x))ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xN

∂u

∂xN+ V (dR (x))

(ϕ′ (dR (x))

ϕ (dR (x))

)2

u

= −V (dR (x))

(∂2u

∂x2N− ϕ′′ (dR (x))

ϕ (dR (x))u

)− V ′ (dR (x))

(∂dR (x)

∂xN

∂u

∂xN− ϕ′ (dR (x))

ϕ (dR (x))u

).

123


Hence, ∫

A(N )2R

u (x)(S+Su

)(x) dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx

+∫

A(N )2R

V ′ (dR (x)) ϕ′ (dR (x)) + V (dR (x)) ϕ′′ (dR (x))

ϕ (dR (x))|u|2 dx .

=∫

A(N )2R

V (dR (x)) |∇u|2 dx −∫

A(N )2R

W (dR (x)) |u|2 dx .


A(N )2R

u (x)(S+Su

)(x) dx

=∫

A(N )2R

|Su|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂u

∂xN− ϕ′ (dR (x))

ϕ (dR (x))

∂dR (x)

∂xNu

∣∣∣∣2

dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂

∂xN

(u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx .

Then we can get

∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx −∫

A(N )R

W (dR (x)) |u|2 dx

=∫

A(N )2R

V (dR (x))

∣∣∣∣ ∂

∂xN

(u

ϕ (dR (x))

)∣∣∣∣2

ϕ2 (dR (x)) dx .

�


For R > 0, we note that

(rα+1,

(α2

)2rα−1 + z20

R2 rα+1


ϕ = r− α2 J0

( r z0R

) = r− α2 J0;R (r). Hence we get the following Hardy type inequality

123



(A(N )2R

):

∫

A(N )2R

|∇u|2 dR (x)α+1 dx −(α

2

)2 ∫

A(N )2R

|u|2 dR (x)α−1 dx

= z20R2

∫

A(N )2R

|u|2 dx +∫

A(N )2R

∣∣∣∣∣∇(

dR (x)α2 u

J0;R(dR (x))

)∣∣∣∣∣2 ∣∣∣∣ J0;R(dR (x))

dR (x)α2

∣∣∣∣2

dx

and ∫

A(N )2R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dR (x)α+1 dx −(α

2

)2 ∫

A(N )2R


= z20R2

∫

A(N )2R

|u|2 dx +∫

A(N )2R

∣∣∣∣∣∂

∂xN

(dR (x)

α2 u

J0;R(dR (x))

)∣∣∣∣∣2 ∣∣∣∣ J0;R(dR (x))

dR (x)α2

∣∣∣∣2

dx .

For any R > 0 and k ∈ N, the pair(rα+1,

(α2

)2rα−1 + rα+1Pk,R (r)

)is a Bessel pair

on (0, R) with ϕ = r− α2

[X1

( rR

) · · · Xk( rR

)]− 12 where

Pk,R (r) = 1

4

1

r2

k∑j=1

X21

( r

R

)· · · X2

j

( r

R

)

X1 (r) = 1

1 − ln r, Xi (r) = X1 (Xi−1 (r)) .

Hence, we deduce

Corollary 3.2 For any R > 0 and k ∈ N, we have∫

A(N )2R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dR (x)α+1 dx −(α

2

)2 ∫

A(N )2R


= 1

4

∫

A(N )2R

|u|2 dR (x)α−1k∑j=1

X21

(dR (x)

R

)· · · X2

j

(dR (x)

R

)dx

+∫

A(N )2R

∣∣∣∣∣∂

∂xN

(dR (x)

α2

[X1

(dR (x)

R

)· · · Xk

(dR (x)

R

)] 12

u

)∣∣∣∣∣2

×

∣∣∣∣∣∣∣∣1

dR (x)α2

[X1

(dR(x)R

)· · · Xk

(dR(x)R

)] 12

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

A(N )2R

|u|2 dR (x)α−1k∑j=1

X21

(dR (x)

R

)· · · X2

j

(dR (x)

R

)dx

123


and

∫

A(N )2R

|∇u|2 dR (x)α+1 dx −(α

2

)2 ∫

A(N )2R


= 1

4

∫

A(N )2R

|u|2 dR (x)α−1k∑j=1

X21

(dR (x)

R

)· · · X2

j

(dR (x)

R

)dx

+∫

A(N )2R

∣∣∣∣∣∇(dR (x)

α2

[X1

(dR (x)

R

)· · · Xk

(dR (x)

R

)] 12

u

)∣∣∣∣∣2

×

∣∣∣∣∣∣∣∣1

dR (x)α2

[X1

(dR(x)R

)· · · Xk

(dR(x)R

)] 12

∣∣∣∣∣∣∣∣

2

dx

≥ 1

4

∫

A(N )2R

|u|2 dR (x)α−1k∑j=1

X21

(dR (x)

R

)· · · X2

j

(dR (x)

R

)dx

Hence, the Hardy inequality with distance to the boundary of the strip A(N )2R can be

improved by adding an infinite number of terms.

4 Hardy identities and inequalities with distance to the hyperplane{xN = 0} on half-spaces

In this section, we will prove the Hardy type identities and inequalities with distance tothe hyperplane {xN = 0}, namely Theorem 1.6. We mention that Hardy-Sobolev-Maz’yainequalities in half spaces have been studied, for example, in [5,8,43,50]. We will also givesome applications of Theorem 1.6.


Proof Let T = √V (xN )∇ − √

V (xN )ϕ′(xN )ϕ(xN )

eN . Then its formal adjoint is T+ =− div(

√V (xN )·) − √


eN ·.Hence,

T+Tu = − div

(√V (xN )

[√V (xN )∇u − √

V (xN )ϕ′ (xN )

ϕ (xN )ueN

])

− √V (xN )

ϕ′ (xN )

ϕ (xN )eN ·

[√V (xN )∇u − √

V (xN )ϕ′ (xN )

ϕ (xN )ueN

]

= − div

(V (xN )

[∇u − ϕ′ (xN )

ϕ (xN )ueN

])

123


− V (xN )ϕ′ (xN )

ϕ (xN )

∂u

∂xN+ V (xN )

(ϕ′ (xN )

ϕ (xN )

)2

u

= −V (xN ) u − V ′ (xN )∂u

∂xN+ V ′ (xN )

ϕ′ (xN )

ϕ (xN )u + V (xN )

ϕ′′ (xN )

ϕ (xN )u.

Thus ∫

A(N )R

u (x)(T+Tu

)(x) dx

=∫

A(N )R

V (xN ) |∇u|2 dx +∫

A(N )R

V ′ (xN ) ϕ′ (xN ) + V (xN ) ϕ′′ (xN )

ϕ (xN )|u|2 dx

=∫

A(N )R

V (xN ) |∇u|2 dx −∫

A(N )R

W (xN ) |u|2 dx .


A(N )R

u (x)(T+Tu

)(x) dx

=∫

A(N )R

|Tu|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣∇(

u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx .

Hence, we deduce∫

A(N )R

V (xN ) |∇u|2 −∫

A(N )R

W (xN ) |u|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣∇(

u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx .

Now, let Su = √V (xN ) ∂u

∂xN− √


u. Then its formal adjoint is S+v =− ∂

∂xN

(√V (xN )v

) − √V (xN )

ϕ′(xN )ϕ(xN )

v.

Therefore,

S+Su = − ∂

∂xN

[√V (xN )

(√V (xN )

∂u

∂xN− √

V (xN )ϕ′ (xN )

ϕ (xN )u

)]

− √V (xN )

ϕ′ (xN )

ϕ (xN )

(√V (xN )

∂u

∂xN− √

V (xN )ϕ′ (xN )

ϕ (xN )u

)

= − ∂

∂xN

(V (xN )

∂u

∂xN− V (xN )

ϕ′ (xN )

ϕ (xN )u

)

123


− V (xN )ϕ′ (xN )

ϕ (xN )

∂u

∂xN+ V (xN )

(ϕ′ (xN )

ϕ (xN )

)2

u

= −V (xN )

(∂2u

∂x2N− ϕ′ (xN )

ϕ (xN )

∂u

∂xN− ϕ′′ (xN ) ϕ (xN ) − (

ϕ′ (xN ))2

ϕ2 (xN )u

)

− V ′ (xN )

(∂u

∂xN− ϕ′ (xN )

ϕ (xN )u

)− V (xN )

ϕ′ (xN )

ϕ (xN )

∂u

∂xN+ V (xN )

(ϕ′ (xN )

ϕ (xN )

)2

u

= −V (xN )

(∂2u

∂x2N− ϕ′′ (xN )

ϕ (xN )u

)− V ′ (xN )

(∂u

∂xN− ϕ′ (xN )

ϕ (xN )u

).

Hence,∫

A(N )R

u (x)(S+Su

)(x) dx

=∫

A(N )R

V (xN )

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx +∫

A(N )R

V ′ (xN ) ϕ′ (xN ) + V (xN ) ϕ′′ (xN )

ϕ (xN )|u|2 dx .

=∫

A(N )R

V (xN ) |∇u|2 dx −∫

A(N )R

W (xN ) |u|2 dx .


A(N )R

u (x)(S+Su

)(x) dx

=∫

A(N )R

|Su|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣ ∂u

∂xN− ϕ′ (xN )

ϕ (xN )u

∣∣∣∣2

dx

=∫

A(N )R

V (xN )

∣∣∣∣ ∂

∂xN

(u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx .

Then we can get

∫

A(N )R

V (xN )

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

dx −∫

A(N )R

W (xN ) |u|2 dx

=∫

A(N )R

V (xN )

∣∣∣∣ ∂

∂xN

(u

ϕ (xN )

)∣∣∣∣2

ϕ2 (xN ) dx .

�

123



For R > 0, we note that

(rα+1,

(α2

)2rα−1 + z20

R2 rα+1


ϕ = r− α2 J0

( r z0R

) = r− α2 J0;R (r). Hence we get the following Hardy type inequality in the

spirit of Brezis-Vázquez and Brezis-Marcus:


(A(N )R

):

∫

A(N )R

|∇u|2 xα+1N dx −

(α

2

)2 ∫

A(N )R

|u|2 xα−1N dx

= z20R2

∫

A(N )R

|u|2 xα+1N dx +

∫

A(N )R

∣∣∣∣∣∣∇⎛⎝ x

α2N u

J0;R(xN )

⎞⎠

∣∣∣∣∣∣2 ∣∣∣∣∣∣

J0;R(xN )

xα2N

∣∣∣∣∣∣2

dx

and

∫

A(N )R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

xα+1N dx −

(α

2

)2 ∫

A(N )R

|u|2 xα−1N dx

= z20R2

∫

A(N )R

|u|2 xα+1N dx +

∫

A(N )R

∣∣∣∣∣∣∂

∂xN

⎛⎝ x

α2N u

J0;R(xN )

⎞⎠

∣∣∣∣∣∣

∣∣∣∣∣∣J0;R(xN )

xα2N

∣∣∣∣∣∣2

dx .

For any R > 0 and k ∈ N, the pair(rα+1,

(α2

)2rα−1 + rα+1Pk,R (r)

)is a Bessel pair

on (0, R) with ϕ = r− α2

[X1

( rR

) · · · Xk( rR

)]− 12 where

Pk,R (r) = 1

4

1

r2

k∑j=1

X21

( r

R

)· · · X2

j

( r

R

)

X1 (r) = 1

1 − ln r, Xi (r) = X1 (Xi−1 (r)) .

Hence, we deduce

Corollary 4.2 For any R > 0 and k ∈ N, we have

∫

A(N )R

|∇u|2 xα+1N dx −

(α

2

)2 ∫

A(N )R

|u|2 xα−1N dx

= 1

4

∫

A(N )R

|u|2 xα−1N

k∑j=1

X21

( xNR

)· · · X2

j

( xNR

)dx

123


+∫

A(N )R

∣∣∣∣∇(x

α2N

[X1

( xNR

)· · · Xk

( xNR

)] 12u

)∣∣∣∣2∣∣∣∣∣∣

1

xα2N

[X1

( xNR

) · · · Xk( xN

R

)] 12

∣∣∣∣∣∣2

dx

≥ 1

4

∫

A(N )R

|u|2 xα−1N

k∑j=1

X21

( xNR

)· · · X2

j

( xNR

)dx

and∫

A(N )R

∣∣∣∣ ∂u

∂xN

∣∣∣∣2

xα+1N dx −

(α

2

)2 ∫

A(N )R

|u|2 xα−1N dx

= 1

4

∫

A(N )R

|u|2 xα−1N

k∑j=1

X21

( xNR

)· · · X2

j

( xNR

)dx

+∫

A(N )R

∣∣∣∣ ∂

∂xN

(x

α2N

[X1

( xNR

)· · · Xk

( xNR

)] 12u

)∣∣∣∣2∣∣∣∣∣∣

1

xα2N

[X1

( xNR

) · · · Xk( xN

R

)] 12

∣∣∣∣∣∣2

dx

≥ 1

4

∫

A(N )R

|u|2 xα−1N

k∑j=1

X21

( xNR

)· · · X2

j

( xNR

)dx .

Hence, the Hardy inequality with distance to the hyperplane {xN = 0} can be improvedby adding an infinite number of terms.

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Affiliations

Nguyen Lam1,2 · Guozhen Lu3 · Lu Zhang4,5

Nguyen [email protected]

1 School of Science & Environment, Grenfell Campus, Memorial University of Newfoundland,Corner Brook, NL A2H 5G4, Canada

123






2 Department of Mathematics, University of British Columbia and The Pacific Institute for theMathematical Sciences, Vancouver, BC V6T1Z4, Canada

3 Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA4 School of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an 710119,

Shaanxi, China5 Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902, USA

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