HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS
FOR BAOUENDI-GRUSHIN VECTOR FIELDS
ISMAIL KOMBE
Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi-
Grushin vector fields : ∇γ = (∇x, |x|
2γ∇y) where γ > 0, ∇x and ∇y are usual gradient
operators in the variables x ∈ Rm and y ∈ Rk, respectively. In the first part of the paper,
we prove some weighted Hardy type inequalities with remainder terms. In the second part,
we prove two versions of weighted Rellich type inequality on the whole space. We find
sharp constants for these inequalities. We also obtain their improved versions for bounded
domains.
1. Introduction
This paper is concerned with Hardy and Rellich type inequalities with remainder terms
for Baouendi-Grushin vector fields. Let x ∈ Rm, y ∈ Rk, γ > 0 and n = m + k, with
m, k ≥ 1. Then the following Hardy type inequality for Baouendi-Grushin vector fields has
been proved by Garofalo [G],
(1.1)
|∇xφ|
2 + |x|2γ |∇yφ|
dxdy ≥
|x|2γ
|x|2+2γ + (1 + γ2)2|y|2
)φ2dxdy
where φ ∈ C∞0 (R
m ×Rk \ {(0, 0)}) and Q = m+(1+ γ)k. Here, ∇xφ and ∇yφ denotes the
gradients of φ in the variables x and y, respectively. A similar inequality with the same
sharp constant (Q−2
)2 holds if Rn replaced by Ω and Ω contains the origin [D]. If γ = 0
then it is clear that the inequality (1.1) recovers the classical Hardy inequality in Rn
(1.2)
|∇φ(z)|2dz ≥
|φ(z)|2
where z = (x, y) ∈ Rm×Rk and the constant (n−2
)2 is sharp. There exists a large literature
concerning with the Hardy inequalities and, in particular, sharp inequalities as well as their
improved versions which have attracted a lot of attention because of their application to
singular problems (See [BG], [PV], [BV], [GP], [CM], [VZ], [K1] and references therein).
A sharp improvement of the Hardy inequality (1.2) was discovered by Brezis and Vázquez
[BV]. They proved that for a bounded domain Ω ⊂ Rn
(1.3)
|∇φ(z)|2dz ≥
|φ(z)|2
dz + µ
φ2dz,
where φ ∈ C∞0 (Ω), ωn and |Ω| denote the n-dimensional Lebesgue measure of the unit
ball B ⊂ Rn and the domain Ω respectively. Here µ is the first eigenvalue of the Laplace
Date: April 09, 2007.
Key words and phrases. Hardy inequality, Rellich inequality, Best constants, Baouendi-Grushin vector
fields.
AMS Subject Classifications: 26D10, 35H20.
http://arxiv.org/abs/0704.1343v1
2 ISMAIL KOMBE
operator in the two dimensional unit disk and it is optimal when Ω is a ball centered at
the origin. In a recent paper Abdelloui, Colorado and Peral [ACP] obtained, among other
things, the following improved Caffarelli-Kohn-Nirenberg inequality
(1.4)
|∇φ(z)|2|z|−2adz ≥
n− 2a− 2
|φ(z)|2
|z|2a+2
dz + C
|∇φ|q|z|−aq
where φ ∈ C∞0 (Ω), −∞ < a <
, 1 < q < 2 and C = C(q, n,Ω) > 0. Motivated
by these results, our first goal is to find improved weighted Hardy type inequalities for
Baouendi-Grushin vector fields.
It is well known that an important extension of Hardy’s inequality to higher-order deriva-
tives is the following Rellich inequality
(1.5)
|∆φ(z)|2dz ≥
n2(n− 4)2
|φ(z)|2
where φ ∈ C∞0 (R
n \ {0}), n 6= 2 and the constant
n2(n−4)2
is sharp. Davies and Hinz [DH],
among other results, obtained sharp weighted Rellich inequalities of the form
(1.6)
|∆φ(z)|2
dz ≥ C
|φ(z)|2
for suitable values of α, β, p and φ ∈ C∞0 (R
n \ {0}). In a recent paper, Tertikas and Zo-
graphopoulos [TZ], among other results, obtained the following new Rellich type inequalities
that connects first to second order derivatives:
(1.7)
|∆φ|2dz ≥
|∇φ|2
where φ ∈ C∞0 (R
n \ {0}) and the constant n
is sharp. Recently, Kombe [K2] obtained
analogues of (1.6) and (1.7), and their improved versions on Carnot groups. Motivated by
the above results, our second goal is to find sharp weighted Rellich type inequalities and
their improved versions for Baouendi-Grushin vector fields in that they do not arise from
any Carnot group. We should also mention that Kombe and Özaydin [KÖ] obtained (under
some geometric assumptions) improved Hardy and Rellich inequalities on a Riemannian
manifold that does not recover our current results. Analogue inequalities for the Greiner
vector fields will be given in a forthcoming paper [K3].
2. Notations and Back ground material
In this section, we shall collect some notations, definitions and preliminary facts which
will be used throughout the article. The generic point is z = (x1, ..., xm, y1, ..., yk) = (x, y) ∈
m × Rk with m, k ≥ 1, m + k = n. The sub-elliptic gradient is the n dimensional vector
field given by
(2.1) ∇γ = (X1, · · · , Xm, Y1, · · · , Yk)
where
(2.2) Xj =
, j = 1, · · · , m, Yj = |x|
, j = 1, · · · , k.
The Baouendi-Grushin operator on Rm+k is the operator
(2.3) ∆γ = ∇γ · ∇γ = ∆x + |x|
2γ∆y,
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS3
where ∆x and ∆y are Laplace operators in the variables x ∈ R
m and y ∈ Rk, respectively
(see [B], [G1], [G2]). If γ is an even positive integer then ∆γ is a sum of squares of C
∞ vector
fields satisfying Hörmander finite rank condition: rank Lie [ X1, · · · , Xm, Y1, · · · , Yk] = n.
The anisotropic dilation attached to ∆γ is given by
δλ(z) = (λx, λ
γ+1y), λ > 0, z = (x, y) ∈ Rm+k.
The change of variable formula for the Lebesgue measure gives that
d ◦ δλ(x, y) = λ
dxdy,
where
Q = m+ (1 + γ)k
is the homogeneous dimension with respect to dilation δλ. For z = (x, y) ∈ R
m × Rk, let
(2.4) ρ = ρ(z) :=
|x|2(1+γ) + (1 + γ)2|y|2
2(1+γ)
By direct computation we get
|∇γρ| =
Let f ∈ C2(0,∞) and define u = f(ρ) then we have the following useful formula
(2.5) ∆γu =
|x|2γ
f ′′ +
We let Bρ = {z ∈ R
n | ρ(z) < r}, Bρ̃ = {z ∈ R
n | ρ̃(z, 0) < r} and call these sets,
respectively, ρ-ball and Carnot-Carathéodory metric ball centered at the origin with radius
r. The Carnot-Carathéodory distance ρ̃ between the points z andz0 is defined by
ρ̃(z, z0) = inf{length(η) | η ∈ K}
where the set K is the set of all curves η such that η(0) = z, η(1) = z0 and η̇(t) is
in span{X1(η(t)), ..., Xm(η(t)), Y1(η(t)), ..., Yk(η(t))}. If γ is a positive even integer then
Carnot-Carathéodory distance of z from the origin ρ̃(z, 0) is comparable to ρ(z). ( See
[FGW] and [Be] for further details.)
It is well known that Sobolev and Poincaré type inequalities are important in the study
of partial differential equations, especially in the study of those arising from geometry
and physics. In[FGW], Franchi, Gutierrez and Wheeden obtained the following Sobolev-
Poincaré inequality for metric balls associated with Baouendi-Grushin type operators:
(2.6)
w1(B)
|∇γφ|
w1(z)dz
w2(B)
|φ(z)|qw2(z)dz
where φ ∈ C∞0 (B) and the weight functions w1 and w2 satisfies some certain conditions.
Here, c is independent of φ and B, 1 ≤ p ≤ q <∞ and w(B) =
w(z)dz. If w1 = w2 = 1
then Monti [M] obtained the following sharp Sobolev inequality
(2.7)
|∇xφ|
2 + |x|2γ|∇yφ|
Q−2dxdy
where C = C(m, k, α) > 0.
4 ISMAIL KOMBE
3. Improved Hardy-type inequalities
In this section we study improved Hardy type inequalities. These inequalities plays key
role in establishing improved Rellich type inequalities. In the various integral inequalities
below (Section 3 and Section 4), we allow the values of the integrals on the left-hand sides
to be +∞. The following theorem is the first result of this section.
Theorem 3.1. Let γ be an even positive integer, α ∈ R, −m
< t < m
, and Q+α− 2 > 0.
Then the following inequality is valid
(3.1)
α|∇γρ|
t|∇γφ|
Q+ α− 2
α |∇γρ|
ρα|∇γρ|
tφ2dz
for all compactly supported smooth function φ ∈ C∞0 (Bρ).
Proof. Let φ = ρβψ ∈ C∞0 (Bρ) and β ∈ R \ {0}. A direct calculation shows that
(3.2)
ρα|∇γρ|
t|∇γφ|
2dz = β2
ρα+2β−2|∇γρ|
t+2ψ2dz
ρα+2β−1|∇γρ|
tψ∇γρ · ∇γψdz
ρα+2β |∇γρ|
t|∇γψ|
Applying integration by parts to the middle term and using the following fact
ρα+2β−1|∇γρ|
= (Q+ α + 2β − 2)ρα+2β−2|∇γρ|
yields
(3.3)
ρα|∇γρ|
t|∇γφ|
2dz = f(β)
ρα+2β−2|∇γρ|
t+2ψ2dz +
ρα+2β |∇γρ|
t|∇γψ|
where f(β) = −β2 − β(α + Q − 2). Note that f(β) attains the maximum for β = 2−α−Q
and this maximum is equal to CH = (
Q+α−2
)2. Therefore we have the following
(3.4)
α|∇γρ|
t|∇γφ|
dz = CH
α−2|∇γρ|
2−Q|∇γρ|
t|∇γψ|
It is easy to show that the weight functions w1 = w2 = ρ
2−Q|∇γρ|
t satisfies the Mucken-
houpt A2 condition for −
< t < m
. Therefore weighted Poincaré inequality holds (see
[FGW], [Lu], [FGaW]) and we have
ρ2−Q|∇γρ|
t|∇γψ|
2dz ≥
ρ2−Q|∇γρ|
tψ2dz
ρα|∇γρ|
tφ2dz
where C is a positive constant and r2 is the radius of the ball Bρ.
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS5
We now obtain the desired inequality
(3.5)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
ρα−2|∇γρ|
t+2φ2dz +
ρα|∇γρ|
tφ2dz.
Using the same method, we have the following weighted Hardy inequality which has a
logarithmic remainder term. Similar results in the Euclidean setting can be found in [FT],
[AR], [WW], [ACP].
Theorem 3.2. Let α ∈ R, t ∈ R, Q+ α− 2 > 0. Then the following inequality is valid
(3.6)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
ρα−2|∇γρ|
t+2φ2dz +
ρα−2|∇γρ|
t+2 φ
(ln r
for all compactly supported smooth function φ ∈ C∞0 (Bρ).
Proof. We have the following result from (3.4):
(3.7)
ρα|∇γρ|
t|∇γφ|
2dz = CH
ρα−2|∇γρ|
t+2φ2dz +
ρ2−Q|∇γρ|
t|∇γψ|
Let ϕ ∈ C∞0 (Bρ) and set ψ(z) = (ln
)1/2ϕ(z). A direct computation shows that
(3.8)
ρ2−Q|∇γρ|
t|∇γψ|
2dz ≥
ρ−Q|∇γρ|
t+2 ψ
(ln r
ρα−2|∇γρ|
t+2 φ
(ln r
Substituting (3.8) into (3.7) which yields the desired inequality (3.6). �
We now first prove the following weighted Lp-Hardy inequality which plays an important
role in the proof of Theorem 3.3, Theorem 4.1 and Theorem 4.5.
Theorem 3.3. Let Ω be either bounded or unbounded domain with smooth boundary which
contains origin, or Rn. Let α ∈ R, t ∈ R, 1 ≤ p < ∞ and Q + α − p > 0. Then the
following inequality holds
(3.9)
ρα|∇γρ|
t|∇γφ|
pdz ≥
Q+ α− p
ρα|∇γρ|
t |∇γρ|
|φ|pdz
for all compactly supported smooth functions φ ∈ C∞0 (Ω).
Proof. Let φ = ρβψ ∈ C∞0 (Ω) and β ∈ R− {0}. We have
|∇γ(ρ
βψ)| = |βρβ−1ψ∇γρ+ ρ
β∇γψ|.
We now use the following inequality which is valid for any a, b ∈ Rn and p > 2,
|a+ b|p − |a|p ≥ c(p)|b|p + p|a|p−2a · b
where c(p) > 0. This yields
ρα|∇γρ|
t|∇φ|p ≥ |β|pρβp−p+α|∇γρ|
p+t|ψ|p + p|β|p−2βρα+βp+1−p|∇γρ|
p+t−2|ψ|p−2ψ∇ρ · ∇ψ.
Integrating over the domain Ω gives
6 ISMAIL KOMBE
(3.10)
ρα|∇γρ|
t|∇φ|pdx ≥ |β|p
ρβp−p+α|∇γρ|
t|ψ|pdz
|β|p−2βρα+βp+1−p|∇γρ|
p+t−2|ψ|p−2ψ∇ρ · ∇ψdz.
Applying integration by parts to second integral on the right-hand side of (3.10) and using
the fact that ∇γ(|∇γρ|) · ∇γρ = 0 then we get
α|∇γρ|
t|∇φ|pdx ≥
|β|p − |β|p−2β(βp− p+ α +Q)
βp−p+α|∇γρ|
p+t|ψ|pdz.
We now choose β = p−Q−α
to get the desired inequality
(3.11)
ρα|∇γρ|
t|∇φ|pdz ≥
Q+ α− p
ρα|∇γρ|
t |∇γρ|
|φ|pdz.
Theorem (3.3) also holds for 1 < p < 2 and in this case we use the following inequality
|a+ b|p − |a|p ≥ c(p)
(|a|+ |b|)2−p
+ p|a|p−2a · b
where c(p) > 0 (see [L]). �
We now have the following improved Hardy inequality which is inspired by recent result
of Abdellaoui, Colorado and Peral [ACP]. It is clear that if γ = t = 0 then our result
recovers the inequality (1.4).
Theorem 3.4. Let Ω ⊂ Rn be a bounded domain with smooth boundary which contains
origin, 1 < q < 2, Q + α − 2 > 0, Q = m + (1 + γ)k and φ ∈ C∞0 (Ω) then there exists a
positive constant C = C(Q, q,Ω) such that the following inequality is valid
(3.12)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
|∇γρ|
φ2dz + C
|∇γφ|
|∇γρ|
where CH =
Q+α−2
Proof. Let φ ∈ C∞0 (Ω) and ψ = ρ
β where β ∈ R \ {0}. Then straightforward computation
shows that
|∇γφ|
2 −∇γ(
) · ∇γψ =
Therefore
|∇γφ|
2 −∇γ(
) · ∇γψ
ρα|∇γρ|
tdz =
ρα|∇γρ|
2 |∇γρ|
where we used the Jensen’s inequality in the last step. Applying integration by parts, we
obtain
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS7
|∇γφ|
2 −∇γ(
) · ∇γψ
ρα|∇γρ|
tdz =
|∇γφ|
2ρα|∇γρ|
α + β
(∆γ(ρ
|∇γρ|
tφ2dz
ρα|∇γρ|
t|∇γφ|
+ β(α+ β +Q− 2)
|∇γρ|
φ2dz.
Therefore we have
(3.13)
ρα|∇γρ|
t|∇γφ|
2dz ≥ −β(α + β +Q− 2)
|∇γρ|
2 |∇γρ|
We can use the following inequality which is valid for any w1, w2 ∈ R
n and 1 < q < 2
(3.14) c(q)|w2|
q ≥ |w1 + w2|
q − |w1|
q − q|w1|
q−2〈w1, w2〉.
Using the inequality (3.14), Young’s inequality and the weighted Lp-Hardy inequality (3.9),
we get
(3.15)
2 |∇γρ|
2 dz ≥ C
|∇γφ|
2 |∇γρ|
where C > 0. Substituting (3.15) into (3.13) then we obtain
ρα|∇γρ|
t|∇γφ|
2dz ≥ −β(α+β+Q−2)
|∇γρ|
φ2dz+C
|∇γφ|
2 |∇γρ|
Now choosing β = 2−α−Q
then we have the following inequality
ρα|∇γρ|
t|∇γφ|
2dz ≥
Q + α− 2
|∇γρ|
φ2dz+C
|∇γφ|
2 |∇γρ|
4. Sharp Weighted Rellich-type inequalities
The main goal of this section is to find sharp analogues of (1.6) and (1.7) for Baouendi-
Grushin vector fields. We then obtain their improved versions for bounded domains. The
proofs are mainly based on Hardy type inequalities. The following is the first result of this
section.
Theorem 4.1. (Rellich type inequality I) Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+ (1 + γ)k
and α > 2. Then the following inequality is valid
(4.1)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
φ2dz.
Moreover, the constant
(Q+α−4)2(Q−α)2
is sharp.
8 ISMAIL KOMBE
Proof. A straightforward computation shows that
(4.2) ∆γρ
α−2 = (Q+ α− 4)(α− 2)ρα−4|∇γρ|
Multiplying both sides of (4.2) by φ2 and integrating over Rn, we obtain
φ2∆γρ
α−2dz =
ρα−2(2φ∆γφ+ 2|∇γφ|
2)dz.
Since
φ2∆γρ
α−2dz = (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz.
Therefore
(4.3) (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdx = 2
ρα−2|∇γφ|
Applying the weighted Hardy inequality (3.9) to the right hand side of (4.3), we get
(4.4) −
ρα−2φ∆γφdz ≥ (
Q+ α− 4
ρα−4|∇γρ|
2φ2dz.
We now apply the Cauchy-Schwarz inequality to obtain
(4.5) −
ρα−2φ∆γφdz ≤
ρα−4|∇γρ|
2φ2dz
)1/2(
|∇γρ|2
|∆γφ|
Substituting (4.5) into (4.4) yields the desired inequality
(4.6)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
φ2dz.
It only remains to show that the constant C(Q,α) =
(Q+α−4)2(Q−α)2
is the best constant
for the Rellich inequality (4.1), that is
(Q+ α− 4)2(Q− α)2
= inf
|∆γf |
|∇γρ|2
|∇γρ|2
f 2dz
, f ∈ C∞0 (R
n), f 6= 0
Given ǫ > 0, take the radial function
(4.7) φǫ(ρ) =
(Q+α−4
+ 1 if ρ ∈ [0, 1],
Q+α−4
+ǫ) if ρ > 1,
where ǫ > 0. In the sequel we indicate B1 = {ρ(z) : ρ(z) ≤ 1} ρ-ball centered at the origin
in Rn with radius 1.
By direct computation we get
(4.8)
|∆γφǫ|
|∇γρ|2
|∆γφǫ|
|∇γρ|2
Bρ\B1
|∆γφǫ|
|∇γρ|2
= A(Q,α, ǫ) +B(Q,α, ǫ)
Bρ\B1
ρ−Q−2ǫ|∇γρ|
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS9
where
B(Q,α, ǫ) = (
Q+ α− 4
+ ǫ)2(
− ǫ)2.
(4.9)
|∇γρ|
φ2dz =
|∇γρ|
φ2dz +
Bρ\B1
|∇γρ|
= C(Q,α, ǫ) +
Bρ\B1
ρ−Q−2ǫdz.
Since Q+α−4 > 0 then A(Q,α, ǫ), and C(Q,α, ǫ) are bounded and we conclude by letting
ǫ −→ 0. �
Using the same argument as above and improved Hardy inequality (3.1), we obtain the
following improved Rellich type inequality.
Theorem 4.2. Let φ ∈ C∞0 (Bρ), Q = m+(1+γ)k and 4−Q < α < Q. Then the following
inequality is valid
(4.10)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
(Q+ α− 4)(Q− α)
2C2r2
ρα−2φ2dz.
Proof. We have the following fact from (4.3):
(4.11) (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdx = 2
ρα−2|∇γφ|
Applying the improved Hardy inequality (3.1) on the right hand side of (4.11), we get
(Q + α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdz
Q+ α− 4
ρα−4|∇γρ|
2φ2dz +
ρα−2φ2dz
Now it is clear that,
(4.12)
ρα−2φ∆γφdz ≥ (
Q + α− 4
ρα−4|∇γρ|
2φ2dz
Next, we apply the Young’s inequality to the expression −
ρα−2φ∆φdz and we obtain
(4.13) −
ρα−2φ∆γφdz ≤ ǫ
ρα−4|∇γρ|
2φ2dz +
|∆γφ|
|∇γρ|2
where ǫ > 0. Combining (4.13) and (4.12), we obtain
|∆γφ|
|∇γρ|2
−4ǫ2− (Q+α−4)(Q−α)ǫ
ρα−4|∇γρ|
2φ2dz+
ρα−2φ2dz.
10 ISMAIL KOMBE
Note that the quadratic function −4ǫ2 − (Q + α − 4)(Q − α)ǫ attains the maximum for
(Q+α−4)(Q−α)
and this maximum is equal to
(Q+α−4)2(Q−α)2
. Therefore we obtain the
desired inequality
(4.14)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
(Q+ α− 4)(Q− α)
2C2r2
ρα−2φ2dz.
Arguing as above, and using the improved Hardy inequalities (3.2) and (3.4) we obtain
the following Rellich type inequalities.
Theorem 4.3. Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then
the following inequality is valid
(4.15)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
ρα−4|∇γρ|
2φ2dz
(Q+ α− 4)(Q− α)
ρα−4|∇γρ|
ln( r
Theorem 4.4. Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then
the following inequality is valid
(4.16)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
C(Q− α)(Q+ 3α− 8)
|∇γφ|
where Ω ⊂ Rn is a bounded domain with smooth boundary.
We now have the following Rellich type inequality that connects first to second order
derivatives. It is clear that if α = γ = 0 then our result covers the inequality (1.7).
Theorem 4.5. (Rellich type inequality II) Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k
and 2 < α < Q. Then the following inequality is valid
(4.17)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
Furthermore, the constant C(Q,α) =
is sharp.
Proof. The proof of this theorem is similar to the proof Theorem (4.1). Using the same
argument as above, we have the following from (4.3)
(4.18) −
ρα−2φ∆γφdx =
ρα−2|∇γφ|
(Q + α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz.
It is clear that (Q+ α− 4)(α− 2) > 0 and using the Hardy inequality (3.9) (p = 2, t = 0)
we get
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS11
(4.19) −
ρα−2φ∆γφdz ≥
Q+ α− 4
ρα−2|∇γφ|
Let us apply Young’s inequality to expression −
ρα−2φ∆γφ dz and we obtain
(4.20)
φ∆γφdz ≤ ǫ
α−4|∇γρ|
α |∆γφ|
|∇γρ|2
Q + α− 4
α−2|∇γφ|
α |∆γφ|
|∇γρ|2
where ǫ > 0 and will be chosen later. Substituting (4.20) into (4.19) and rearranging terms,
we get
(4.21)
|∆γφ|
|∇γρ|2
−16ǫ2
(Q+ α− 4)2
( Q− α
Q+ α− 4
|∇γφ|
Choosing ǫ = 1
(Q− α)(Q+ α− 4) which yields the desired inequality
(4.22)
α |∆γφ|
|∇γρ|2
(Q− α)2
α |∇γφ|
To show that constant
is sharp, we use the same sequence of functions (4.7) and
we get
|∆γφǫ|
|∇γρ|2
|∇γφǫ|2
(Q− α
as ǫ −→ 0.
Now, using the same argument as above and improved Hardy inequalities (3.1), (3.6)
and (3.7) we obtain the following improved Rellich type inequalities.
Theorem 4.6. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following
inequality is valid
(4.23)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
(Q− α)(Q+ 3α− 8)
4C2r2
where C > 0 and r is the radius of the ball Bρ.
Theorem 4.7. Let Ω be a bounded domain with smooth boundary ∂Ω. Let φ ∈ C∞0 (Ω),
Q = m+ (1 + γ)k and 2 < α < Q. Then the following inequality is valid
(4.24)
|∆γφ|
|∇γρ|2
dz ≥ (
|∇γφ|
dz + C̃
|∇γφ|
q(α−2)
where C̃ =
C(Q−α)(Q+3α−8)
and C > 0.
Theorem 4.8. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following
inequality is valid
(4.25)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
dz + C(Q,α)
ρα−4|∇γρ|
(ln r
where C(Q,α) =
(Q−α)(Q+3α−8)
12 ISMAIL KOMBE
References
[ACP] B. Abdellaoui, D. Colorado, I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc.
Var. Partial Differential Equations 23 (2005), no. 3, 327-345.
[AR] Adimurthi N. Chaudhuri and M. Ramaswamy, An improved HardySobolev inequality and its applica-
tions, Proc. Amer. Math. Soc. 130 (2002), pp. 489505.
[BG] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc.
284 (1984), 121-139.
[B] M. S. Baouendi, Sur une classe d’opérateurs elliptiques dégénérés, Bull. Soc. Math. France 95, 45-87
(1967).
[Be] Belläıche, André. The Tangent Space in Sub-Riemannian Geometry, 1–78, Progr. Math., 144,
Birkhuser, Basel, 1996.
[BV] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ.
Complutense Madrid 10 (1997), 443-469.
[CM] X. Cabré and Y. Martel, Existence versus explosion instantane pour des quations de la chaleur
linaires avec potentiel singulier , C. R. Acad. Sci. Paris Sr. I. Math., 329 (1999), 973-978.
[D] L. D’Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132
(2004), no. 3, 725-734.
[DH] E. B. Davies, and A. M. Hinz, Explicit constants for Rellich inequalities in Lp(Ω), Math. Z. 227
(1998), no. 3, 511-523.
[FGW] B. Franchi, C. E. Gutirrez, and R. L. Wheeden, Weighted Sobolev-Poincar inequalities for Grushin
type operators, Comm. Partial Differential Equations 19, 523-604 (1994).
[FGaW] B. Franchi, S. Gallot and R. L. Wheeden Sobolev and isoperimetric inequalities for degenerate
metrics, Math. Ann. 300 (1994), no. 4, 557-571.
[FT] S. Filippas and A. Tertikas Optimizing Improved Hardy Inequalities, J. Funct. Anal. 192, (2002),186-
[G] N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of
arbitrary codimension, J. Differential Equations 104 (1993), no. 1, 117-146.
[GP] J. Garcia Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems,
J. Diff. Equations, 144 (1998), 441-476.
[GK] J. A. Goldstein and I. Kombe, Nonlinear parabolic differential equations with the singular lower order
term, Adv. Differential Equations 10 (2003), 1153-1192.
[G1] V. Grushin, A certain class of hypoelliptic operators, Math. USSR-Sb. 12, No. 3, 458-476 (1970)
[G2] V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submani-
fold, Mat. Sb. 84, 163-195 (1971).
[K1] I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators, Math. Nachr.
279 (2006), no. 7, 756-773.
[K2] I. Kombe, Hardy, Rellich and Uncertainty principle inequalities on Carnot Groups, preprint.
[K3] I. Kombe, Sharp Hardy and Rellich type inequalites with remainders for the Greiner vector fields,
preprint.
[KÖ] I. Kombe and M. Özaydin Improved Hardy and Rellich inequalities on Riemannian manifolds,
preprint.
[L] P. Lindqvist, On the equation div(|∇u|p−2∇u) + λ|u|p−2u = 0, Proc. Amer. Math. Soc. (109) (1990),
157-164.
[Lu] G. Lu, Weighted Poincar and Sobolev inequalities for vector fields satisfying Hrmander’s condition
and applications, Rev. Mat. Iberoamericana 8 (1992), no. 3, 367-439.
[M] R. Monti Sobolev inequalities for weighted gradients, Comm. Partial Differential Equations 31 (2006),
no. 10-12, 1479-1504.
[PV] I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear
heat equation with exponential reaction term, Arch. Rational Mech. Anal. 129 (1995), 201-224.
[TZ] A. Tertikas and N. Zographopoulos, Best constants in the Hardy-Rellich Inequalities and Related
Improvements, Adv. Math. 209,2 (2007), 407-459.
[VZ] J. L. Vázquez and E. Zuazua, The Hardy constant and the asymptotic behaviour of the heat equation
with an inverse-square potential , J. Funct. Anal. 173 (2000), 103-153.
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS13
[WW] Z.-Q. Wang and M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct.
Anal. 203 (2003), no. 2, 550-568.
Ismail Kombe, Mathematics Department, Dawson-Loeffler Science &Mathematics Bldg,
Oklahoma City University, 2501 N. Blackwelder, Oklahoma City, OK 73106-1493
E-mail address : ikombe@okcu.edu
	1. Introduction
	2. Notations and Back ground material
	3. Improved Hardy-type inequalities
	4. Sharp Weighted Rellich-type inequalities
	References

  In this paper we study Hardy and Rellich type inequalities for
Baouendi-Grushin vector fields : $\nabla_{\gamma}=(\nabla_x,
|x|^{2\gamma}\nabla_y)$ where $\gamma>0$, $\nabla_x$ and $\nabla_y$ are usual
gradient operators in the variables $x\in \mathbb{R}^m$ and $y\in\mathbb{R}^k$,
respectively. In the first part of the paper, we prove some weighted Hardy type
inequalities with remainder terms. In the second part, we prove two versions of
weighted Rellich type inequality on the whole space. We find sharp constants
for these inequalities. We also obtain their improved versions for bounded
domains.

Introduction
This paper is concerned with Hardy and Rellich type inequalities with remainder terms
for Baouendi-Grushin vector fields. Let x ∈ Rm, y ∈ Rk, γ > 0 and n = m + k, with
m, k ≥ 1. Then the following Hardy type inequality for Baouendi-Grushin vector fields has
been proved by Garofalo [G],
(1.1)
|∇xφ|
2 + |x|2γ |∇yφ|
dxdy ≥
|x|2γ
|x|2+2γ + (1 + γ2)2|y|2
)φ2dxdy
where φ ∈ C∞0 (R
m ×Rk \ {(0, 0)}) and Q = m+(1+ γ)k. Here, ∇xφ and ∇yφ denotes the
gradients of φ in the variables x and y, respectively. A similar inequality with the same
sharp constant (Q−2
)2 holds if Rn replaced by Ω and Ω contains the origin [D]. If γ = 0
then it is clear that the inequality (1.1) recovers the classical Hardy inequality in Rn
(1.2)
|∇φ(z)|2dz ≥
|φ(z)|2
where z = (x, y) ∈ Rm×Rk and the constant (n−2
)2 is sharp. There exists a large literature
concerning with the Hardy inequalities and, in particular, sharp inequalities as well as their
improved versions which have attracted a lot of attention because of their application to
singular problems (See [BG], [PV], [BV], [GP], [CM], [VZ], [K1] and references therein).
A sharp improvement of the Hardy inequality (1.2) was discovered by Brezis and Vázquez
[BV]. They proved that for a bounded domain Ω ⊂ Rn
(1.3)
|∇φ(z)|2dz ≥
|φ(z)|2
dz + µ
φ2dz,
where φ ∈ C∞0 (Ω), ωn and |Ω| denote the n-dimensional Lebesgue measure of the unit
ball B ⊂ Rn and the domain Ω respectively. Here µ is the first eigenvalue of the Laplace
Date: April 09, 2007.
Key words and phrases. Hardy inequality, Rellich inequality, Best constants, Baouendi-Grushin vector
fields.
AMS Subject Classifications: 26D10, 35H20.
http://arxiv.org/abs/0704.1343v1
2 ISMAIL KOMBE
operator in the two dimensional unit disk and it is optimal when Ω is a ball centered at
the origin. In a recent paper Abdelloui, Colorado and Peral [ACP] obtained, among other
things, the following improved Caffarelli-Kohn-Nirenberg inequality
(1.4)
|∇φ(z)|2|z|−2adz ≥
n− 2a− 2
|φ(z)|2
|z|2a+2
dz + C
|∇φ|q|z|−aq
where φ ∈ C∞0 (Ω), −∞ < a <
, 1 < q < 2 and C = C(q, n,Ω) > 0. Motivated
by these results, our first goal is to find improved weighted Hardy type inequalities for
Baouendi-Grushin vector fields.
It is well known that an important extension of Hardy’s inequality to higher-order deriva-
tives is the following Rellich inequality
(1.5)
|∆φ(z)|2dz ≥
n2(n− 4)2
|φ(z)|2
where φ ∈ C∞0 (R
n \ {0}), n 6= 2 and the constant
n2(n−4)2
is sharp. Davies and Hinz [DH],
among other results, obtained sharp weighted Rellich inequalities of the form
(1.6)
|∆φ(z)|2
dz ≥ C
|φ(z)|2
for suitable values of α, β, p and φ ∈ C∞0 (R
n \ {0}). In a recent paper, Tertikas and Zo-
graphopoulos [TZ], among other results, obtained the following new Rellich type inequalities
that connects first to second order derivatives:
(1.7)
|∆φ|2dz ≥
|∇φ|2
where φ ∈ C∞0 (R
n \ {0}) and the constant n
is sharp. Recently, Kombe [K2] obtained
analogues of (1.6) and (1.7), and their improved versions on Carnot groups. Motivated by
the above results, our second goal is to find sharp weighted Rellich type inequalities and
their improved versions for Baouendi-Grushin vector fields in that they do not arise from
any Carnot group. We should also mention that Kombe and Özaydin [KÖ] obtained (under
some geometric assumptions) improved Hardy and Rellich inequalities on a Riemannian
manifold that does not recover our current results. Analogue inequalities for the Greiner
vector fields will be given in a forthcoming paper [K3].
2. Notations and Back ground material
In this section, we shall collect some notations, definitions and preliminary facts which
will be used throughout the article. The generic point is z = (x1, ..., xm, y1, ..., yk) = (x, y) ∈
m × Rk with m, k ≥ 1, m + k = n. The sub-elliptic gradient is the n dimensional vector
field given by
(2.1) ∇γ = (X1, · · · , Xm, Y1, · · · , Yk)
where
(2.2) Xj =
, j = 1, · · · , m, Yj = |x|
, j = 1, · · · , k.
The Baouendi-Grushin operator on Rm+k is the operator
(2.3) ∆γ = ∇γ · ∇γ = ∆x + |x|
2γ∆y,
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS3
where ∆x and ∆y are Laplace operators in the variables x ∈ R
m and y ∈ Rk, respectively
(see [B], [G1], [G2]). If γ is an even positive integer then ∆γ is a sum of squares of C
∞ vector
fields satisfying Hörmander finite rank condition: rank Lie [ X1, · · · , Xm, Y1, · · · , Yk] = n.
The anisotropic dilation attached to ∆γ is given by
δλ(z) = (λx, λ
γ+1y), λ > 0, z = (x, y) ∈ Rm+k.
The change of variable formula for the Lebesgue measure gives that
d ◦ δλ(x, y) = λ
dxdy,
where
Q = m+ (1 + γ)k
is the homogeneous dimension with respect to dilation δλ. For z = (x, y) ∈ R
m × Rk, let
(2.4) ρ = ρ(z) :=
|x|2(1+γ) + (1 + γ)2|y|2
2(1+γ)
By direct computation we get
|∇γρ| =
Let f ∈ C2(0,∞) and define u = f(ρ) then we have the following useful formula
(2.5) ∆γu =
|x|2γ
f ′′ +
We let Bρ = {z ∈ R
n | ρ(z) < r}, Bρ̃ = {z ∈ R
n | ρ̃(z, 0) < r} and call these sets,
respectively, ρ-ball and Carnot-Carathéodory metric ball centered at the origin with radius
r. The Carnot-Carathéodory distance ρ̃ between the points z andz0 is defined by
ρ̃(z, z0) = inf{length(η) | η ∈ K}
where the set K is the set of all curves η such that η(0) = z, η(1) = z0 and η̇(t) is
in span{X1(η(t)), ..., Xm(η(t)), Y1(η(t)), ..., Yk(η(t))}. If γ is a positive even integer then
Carnot-Carathéodory distance of z from the origin ρ̃(z, 0) is comparable to ρ(z). ( See
[FGW] and [Be] for further details.)
It is well known that Sobolev and Poincaré type inequalities are important in the study
of partial differential equations, especially in the study of those arising from geometry
and physics. In[FGW], Franchi, Gutierrez and Wheeden obtained the following Sobolev-
Poincaré inequality for metric balls associated with Baouendi-Grushin type operators:
(2.6)
w1(B)
|∇γφ|
w1(z)dz
w2(B)
|φ(z)|qw2(z)dz
where φ ∈ C∞0 (B) and the weight functions w1 and w2 satisfies some certain conditions.
Here, c is independent of φ and B, 1 ≤ p ≤ q <∞ and w(B) =
w(z)dz. If w1 = w2 = 1
then Monti [M] obtained the following sharp Sobolev inequality
(2.7)
|∇xφ|
2 + |x|2γ|∇yφ|
Q−2dxdy
where C = C(m, k, α) > 0.
4 ISMAIL KOMBE
3. Improved Hardy-type inequalities
In this section we study improved Hardy type inequalities. These inequalities plays key
role in establishing improved Rellich type inequalities. In the various integral inequalities
below (Section 3 and Section 4), we allow the values of the integrals on the left-hand sides
to be +∞. The following theorem is the first result of this section.
Theorem 3.1. Let γ be an even positive integer, α ∈ R, −m
< t < m
, and Q+α− 2 > 0.
Then the following inequality is valid
(3.1)
α|∇γρ|
t|∇γφ|
Q+ α− 2
α |∇γρ|
ρα|∇γρ|
tφ2dz
for all compactly supported smooth function φ ∈ C∞0 (Bρ).
Proof. Let φ = ρβψ ∈ C∞0 (Bρ) and β ∈ R \ {0}. A direct calculation shows that
(3.2)
ρα|∇γρ|
t|∇γφ|
2dz = β2
ρα+2β−2|∇γρ|
t+2ψ2dz
ρα+2β−1|∇γρ|
tψ∇γρ · ∇γψdz
ρα+2β |∇γρ|
t|∇γψ|
Applying integration by parts to the middle term and using the following fact
ρα+2β−1|∇γρ|
= (Q+ α + 2β − 2)ρα+2β−2|∇γρ|
yields
(3.3)
ρα|∇γρ|
t|∇γφ|
2dz = f(β)
ρα+2β−2|∇γρ|
t+2ψ2dz +
ρα+2β |∇γρ|
t|∇γψ|
where f(β) = −β2 − β(α + Q − 2). Note that f(β) attains the maximum for β = 2−α−Q
and this maximum is equal to CH = (
Q+α−2
)2. Therefore we have the following
(3.4)
α|∇γρ|
t|∇γφ|
dz = CH
α−2|∇γρ|
2−Q|∇γρ|
t|∇γψ|
It is easy to show that the weight functions w1 = w2 = ρ
2−Q|∇γρ|
t satisfies the Mucken-
houpt A2 condition for −
< t < m
. Therefore weighted Poincaré inequality holds (see
[FGW], [Lu], [FGaW]) and we have
ρ2−Q|∇γρ|
t|∇γψ|
2dz ≥
ρ2−Q|∇γρ|
tψ2dz
ρα|∇γρ|
tφ2dz
where C is a positive constant and r2 is the radius of the ball Bρ.
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS5
We now obtain the desired inequality
(3.5)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
ρα−2|∇γρ|
t+2φ2dz +
ρα|∇γρ|
tφ2dz.
Using the same method, we have the following weighted Hardy inequality which has a
logarithmic remainder term. Similar results in the Euclidean setting can be found in [FT],
[AR], [WW], [ACP].
Theorem 3.2. Let α ∈ R, t ∈ R, Q+ α− 2 > 0. Then the following inequality is valid
(3.6)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
ρα−2|∇γρ|
t+2φ2dz +
ρα−2|∇γρ|
t+2 φ
(ln r
for all compactly supported smooth function φ ∈ C∞0 (Bρ).
Proof. We have the following result from (3.4):
(3.7)
ρα|∇γρ|
t|∇γφ|
2dz = CH
ρα−2|∇γρ|
t+2φ2dz +
ρ2−Q|∇γρ|
t|∇γψ|
Let ϕ ∈ C∞0 (Bρ) and set ψ(z) = (ln
)1/2ϕ(z). A direct computation shows that
(3.8)
ρ2−Q|∇γρ|
t|∇γψ|
2dz ≥
ρ−Q|∇γρ|
t+2 ψ
(ln r
ρα−2|∇γρ|
t+2 φ
(ln r
Substituting (3.8) into (3.7) which yields the desired inequality (3.6). �
We now first prove the following weighted Lp-Hardy inequality which plays an important
role in the proof of Theorem 3.3, Theorem 4.1 and Theorem 4.5.
Theorem 3.3. Let Ω be either bounded or unbounded domain with smooth boundary which
contains origin, or Rn. Let α ∈ R, t ∈ R, 1 ≤ p < ∞ and Q + α − p > 0. Then the
following inequality holds
(3.9)
ρα|∇γρ|
t|∇γφ|
pdz ≥
Q+ α− p
ρα|∇γρ|
t |∇γρ|
|φ|pdz
for all compactly supported smooth functions φ ∈ C∞0 (Ω).
Proof. Let φ = ρβψ ∈ C∞0 (Ω) and β ∈ R− {0}. We have
|∇γ(ρ
βψ)| = |βρβ−1ψ∇γρ+ ρ
β∇γψ|.
We now use the following inequality which is valid for any a, b ∈ Rn and p > 2,
|a+ b|p − |a|p ≥ c(p)|b|p + p|a|p−2a · b
where c(p) > 0. This yields
ρα|∇γρ|
t|∇φ|p ≥ |β|pρβp−p+α|∇γρ|
p+t|ψ|p + p|β|p−2βρα+βp+1−p|∇γρ|
p+t−2|ψ|p−2ψ∇ρ · ∇ψ.
Integrating over the domain Ω gives
6 ISMAIL KOMBE
(3.10)
ρα|∇γρ|
t|∇φ|pdx ≥ |β|p
ρβp−p+α|∇γρ|
t|ψ|pdz
|β|p−2βρα+βp+1−p|∇γρ|
p+t−2|ψ|p−2ψ∇ρ · ∇ψdz.
Applying integration by parts to second integral on the right-hand side of (3.10) and using
the fact that ∇γ(|∇γρ|) · ∇γρ = 0 then we get
α|∇γρ|
t|∇φ|pdx ≥
|β|p − |β|p−2β(βp− p+ α +Q)
βp−p+α|∇γρ|
p+t|ψ|pdz.
We now choose β = p−Q−α
to get the desired inequality
(3.11)
ρα|∇γρ|
t|∇φ|pdz ≥
Q+ α− p
ρα|∇γρ|
t |∇γρ|
|φ|pdz.
Theorem (3.3) also holds for 1 < p < 2 and in this case we use the following inequality
|a+ b|p − |a|p ≥ c(p)
(|a|+ |b|)2−p
+ p|a|p−2a · b
where c(p) > 0 (see [L]). �
We now have the following improved Hardy inequality which is inspired by recent result
of Abdellaoui, Colorado and Peral [ACP]. It is clear that if γ = t = 0 then our result
recovers the inequality (1.4).
Theorem 3.4. Let Ω ⊂ Rn be a bounded domain with smooth boundary which contains
origin, 1 < q < 2, Q + α − 2 > 0, Q = m + (1 + γ)k and φ ∈ C∞0 (Ω) then there exists a
positive constant C = C(Q, q,Ω) such that the following inequality is valid
(3.12)
ρα|∇γρ|
t|∇γφ|
2dz ≥ CH
|∇γρ|
φ2dz + C
|∇γφ|
|∇γρ|
where CH =
Q+α−2
Proof. Let φ ∈ C∞0 (Ω) and ψ = ρ
β where β ∈ R \ {0}. Then straightforward computation
shows that
|∇γφ|
2 −∇γ(
) · ∇γψ =
Therefore
|∇γφ|
2 −∇γ(
) · ∇γψ
ρα|∇γρ|
tdz =
ρα|∇γρ|
2 |∇γρ|
where we used the Jensen’s inequality in the last step. Applying integration by parts, we
obtain
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS7
|∇γφ|
2 −∇γ(
) · ∇γψ
ρα|∇γρ|
tdz =
|∇γφ|
2ρα|∇γρ|
α + β
(∆γ(ρ
|∇γρ|
tφ2dz
ρα|∇γρ|
t|∇γφ|
+ β(α+ β +Q− 2)
|∇γρ|
φ2dz.
Therefore we have
(3.13)
ρα|∇γρ|
t|∇γφ|
2dz ≥ −β(α + β +Q− 2)
|∇γρ|
2 |∇γρ|
We can use the following inequality which is valid for any w1, w2 ∈ R
n and 1 < q < 2
(3.14) c(q)|w2|
q ≥ |w1 + w2|
q − |w1|
q − q|w1|
q−2〈w1, w2〉.
Using the inequality (3.14), Young’s inequality and the weighted Lp-Hardy inequality (3.9),
we get
(3.15)
2 |∇γρ|
2 dz ≥ C
|∇γφ|
2 |∇γρ|
where C > 0. Substituting (3.15) into (3.13) then we obtain
ρα|∇γρ|
t|∇γφ|
2dz ≥ −β(α+β+Q−2)
|∇γρ|
φ2dz+C
|∇γφ|
2 |∇γρ|
Now choosing β = 2−α−Q
then we have the following inequality
ρα|∇γρ|
t|∇γφ|
2dz ≥
Q + α− 2
|∇γρ|
φ2dz+C
|∇γφ|
2 |∇γρ|
4. Sharp Weighted Rellich-type inequalities
The main goal of this section is to find sharp analogues of (1.6) and (1.7) for Baouendi-
Grushin vector fields. We then obtain their improved versions for bounded domains. The
proofs are mainly based on Hardy type inequalities. The following is the first result of this
section.
Theorem 4.1. (Rellich type inequality I) Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+ (1 + γ)k
and α > 2. Then the following inequality is valid
(4.1)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
φ2dz.
Moreover, the constant
(Q+α−4)2(Q−α)2
is sharp.
8 ISMAIL KOMBE
Proof. A straightforward computation shows that
(4.2) ∆γρ
α−2 = (Q+ α− 4)(α− 2)ρα−4|∇γρ|
Multiplying both sides of (4.2) by φ2 and integrating over Rn, we obtain
φ2∆γρ
α−2dz =
ρα−2(2φ∆γφ+ 2|∇γφ|
2)dz.
Since
φ2∆γρ
α−2dz = (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz.
Therefore
(4.3) (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdx = 2
ρα−2|∇γφ|
Applying the weighted Hardy inequality (3.9) to the right hand side of (4.3), we get
(4.4) −
ρα−2φ∆γφdz ≥ (
Q+ α− 4
ρα−4|∇γρ|
2φ2dz.
We now apply the Cauchy-Schwarz inequality to obtain
(4.5) −
ρα−2φ∆γφdz ≤
ρα−4|∇γρ|
2φ2dz
)1/2(
|∇γρ|2
|∆γφ|
Substituting (4.5) into (4.4) yields the desired inequality
(4.6)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
φ2dz.
It only remains to show that the constant C(Q,α) =
(Q+α−4)2(Q−α)2
is the best constant
for the Rellich inequality (4.1), that is
(Q+ α− 4)2(Q− α)2
= inf
|∆γf |
|∇γρ|2
|∇γρ|2
f 2dz
, f ∈ C∞0 (R
n), f 6= 0
Given ǫ > 0, take the radial function
(4.7) φǫ(ρ) =
(Q+α−4
+ 1 if ρ ∈ [0, 1],
Q+α−4
+ǫ) if ρ > 1,
where ǫ > 0. In the sequel we indicate B1 = {ρ(z) : ρ(z) ≤ 1} ρ-ball centered at the origin
in Rn with radius 1.
By direct computation we get
(4.8)
|∆γφǫ|
|∇γρ|2
|∆γφǫ|
|∇γρ|2
Bρ\B1
|∆γφǫ|
|∇γρ|2
= A(Q,α, ǫ) +B(Q,α, ǫ)
Bρ\B1
ρ−Q−2ǫ|∇γρ|
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS9
where
B(Q,α, ǫ) = (
Q+ α− 4
+ ǫ)2(
− ǫ)2.
(4.9)
|∇γρ|
φ2dz =
|∇γρ|
φ2dz +
Bρ\B1
|∇γρ|
= C(Q,α, ǫ) +
Bρ\B1
ρ−Q−2ǫdz.
Since Q+α−4 > 0 then A(Q,α, ǫ), and C(Q,α, ǫ) are bounded and we conclude by letting
ǫ −→ 0. �
Using the same argument as above and improved Hardy inequality (3.1), we obtain the
following improved Rellich type inequality.
Theorem 4.2. Let φ ∈ C∞0 (Bρ), Q = m+(1+γ)k and 4−Q < α < Q. Then the following
inequality is valid
(4.10)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
(Q+ α− 4)(Q− α)
2C2r2
ρα−2φ2dz.
Proof. We have the following fact from (4.3):
(4.11) (Q+ α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdx = 2
ρα−2|∇γφ|
Applying the improved Hardy inequality (3.1) on the right hand side of (4.11), we get
(Q + α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz − 2
ρα−2φ∆γφdz
Q+ α− 4
ρα−4|∇γρ|
2φ2dz +
ρα−2φ2dz
Now it is clear that,
(4.12)
ρα−2φ∆γφdz ≥ (
Q + α− 4
ρα−4|∇γρ|
2φ2dz
Next, we apply the Young’s inequality to the expression −
ρα−2φ∆φdz and we obtain
(4.13) −
ρα−2φ∆γφdz ≤ ǫ
ρα−4|∇γρ|
2φ2dz +
|∆γφ|
|∇γρ|2
where ǫ > 0. Combining (4.13) and (4.12), we obtain
|∆γφ|
|∇γρ|2
−4ǫ2− (Q+α−4)(Q−α)ǫ
ρα−4|∇γρ|
2φ2dz+
ρα−2φ2dz.
10 ISMAIL KOMBE
Note that the quadratic function −4ǫ2 − (Q + α − 4)(Q − α)ǫ attains the maximum for
(Q+α−4)(Q−α)
and this maximum is equal to
(Q+α−4)2(Q−α)2
. Therefore we obtain the
desired inequality
(4.14)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
(Q+ α− 4)(Q− α)
2C2r2
ρα−2φ2dz.
Arguing as above, and using the improved Hardy inequalities (3.2) and (3.4) we obtain
the following Rellich type inequalities.
Theorem 4.3. Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then
the following inequality is valid
(4.15)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
ρα−4|∇γρ|
2φ2dz
(Q+ α− 4)(Q− α)
ρα−4|∇γρ|
ln( r
Theorem 4.4. Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k and 4−Q < α < Q. Then
the following inequality is valid
(4.16)
|∇γρ|2
|∆γφ|
2dz ≥
(Q+ α− 4)2(Q− α)2
|∇γρ|
C(Q− α)(Q+ 3α− 8)
|∇γφ|
where Ω ⊂ Rn is a bounded domain with smooth boundary.
We now have the following Rellich type inequality that connects first to second order
derivatives. It is clear that if α = γ = 0 then our result covers the inequality (1.7).
Theorem 4.5. (Rellich type inequality II) Let φ ∈ C∞0 (R
m+k \ {(0, 0)}), Q = m+(1+ γ)k
and 2 < α < Q. Then the following inequality is valid
(4.17)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
Furthermore, the constant C(Q,α) =
is sharp.
Proof. The proof of this theorem is similar to the proof Theorem (4.1). Using the same
argument as above, we have the following from (4.3)
(4.18) −
ρα−2φ∆γφdx =
ρα−2|∇γφ|
(Q + α− 4)(α− 2)
ρα−4|∇γρ|
2φ2dz.
It is clear that (Q+ α− 4)(α− 2) > 0 and using the Hardy inequality (3.9) (p = 2, t = 0)
we get
HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS11
(4.19) −
ρα−2φ∆γφdz ≥
Q+ α− 4
ρα−2|∇γφ|
Let us apply Young’s inequality to expression −
ρα−2φ∆γφ dz and we obtain
(4.20)
φ∆γφdz ≤ ǫ
α−4|∇γρ|
α |∆γφ|
|∇γρ|2
Q + α− 4
α−2|∇γφ|
α |∆γφ|
|∇γρ|2
where ǫ > 0 and will be chosen later. Substituting (4.20) into (4.19) and rearranging terms,
we get
(4.21)
|∆γφ|
|∇γρ|2
−16ǫ2
(Q+ α− 4)2
( Q− α
Q+ α− 4
|∇γφ|
Choosing ǫ = 1
(Q− α)(Q+ α− 4) which yields the desired inequality
(4.22)
α |∆γφ|
|∇γρ|2
(Q− α)2
α |∇γφ|
To show that constant
is sharp, we use the same sequence of functions (4.7) and
we get
|∆γφǫ|
|∇γρ|2
|∇γφǫ|2
(Q− α
as ǫ −→ 0.
Now, using the same argument as above and improved Hardy inequalities (3.1), (3.6)
and (3.7) we obtain the following improved Rellich type inequalities.
Theorem 4.6. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following
inequality is valid
(4.23)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
(Q− α)(Q+ 3α− 8)
4C2r2
where C > 0 and r is the radius of the ball Bρ.
Theorem 4.7. Let Ω be a bounded domain with smooth boundary ∂Ω. Let φ ∈ C∞0 (Ω),
Q = m+ (1 + γ)k and 2 < α < Q. Then the following inequality is valid
(4.24)
|∆γφ|
|∇γρ|2
dz ≥ (
|∇γφ|
dz + C̃
|∇γφ|
q(α−2)
where C̃ =
C(Q−α)(Q+3α−8)
and C > 0.
Theorem 4.8. Let φ ∈ C∞0 (Bρ), Q = m + (1 + γ)k and 2 < α < Q. Then the following
inequality is valid
(4.25)
|∆γφ|
|∇γρ|2
(Q− α)2
|∇γφ|
dz + C(Q,α)
ρα−4|∇γρ|
(ln r
where C(Q,α) =
(Q−α)(Q+3α−8)
12 ISMAIL KOMBE
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Ismail Kombe, Mathematics Department, Dawson-Loeffler Science &Mathematics Bldg,
Oklahoma City University, 2501 N. Blackwelder, Oklahoma City, OK 73106-1493
E-mail address : ikombe@okcu.edu
	1. Introduction
	2. Notations and Back ground material
	3. Improved Hardy-type inequalities
	4. Sharp Weighted Rellich-type inequalities
	References
