An improved Hardy type inequality on Heisenberg group
Journal of Inequalities and Applications
An improved Hardy type inequality on Heisenberg group
Ying-Xiong Xiao yxxiao2011@163 0
0 com School of Mathematics and Statistics, Xiaogan University , Xiaogan, Hubei, 432000 , People's Republic of China
Motivated by the work of Ghoussoub and Moradifam, we prove some improved Hardy inequalities on the Heisenberg group Hn via Bessel function. Mathematics Subject Classification (2000): Primary 26D10 and (N − 2)2 is the best constant in (1.1) and is never achieved. A similar inequality 4 with the same best constant holds in ℝN is replaced by an arbitrary domain Ω ⊂ ℝN and Ω contains the origin. Moreover, in case Ω ⊂ ℝN is a bounded domain, Brezis and Vázquez [1] have improved it by establishing that for u ∈ C0∞( ), where ωN and |Ω| denote the volume of the unit ball and Ω, respectively, and z0 = 2.4048... denotes the first zero of the Bessel function J0(z). Inequality (1.2) is optimal in case Ω is a ball centered at zero. Triggered by the work of Brezis and Vázquez (1.2), several Hardy inequalities have been established in recent years. In particular, Adimurthi et al.([2]) proved that, for u ∈ C0∞( ), there exists a constant Cn,k such that
Hardy inequality; Heisenberg group
-
|∇u|2dx ≥
|∇u|2dx ≥
|∇u|2dx ≥
|∇u|2 ≥
|∇u|2dx ≥
if and only if the ordinary differential equation
has a positive solution on (0, r]. These include inequalities (1.2)-(1.4).
Motivated by the work of Ghoussoub and Moradifam ([4]), in this note, we shall
prove similar improved Hardy inequality on the Heisenberg group Hn. Recall that the
Heisenberg group Hn is the Carnot group of step two whose group structure is given
by
⎛
(x, t) ◦ (x , t ) = ⎝ x + x , t + t + 2
The vector fields
X2j−1 =
∂ ∂
∂x2j−1 + 2x2j ∂t ,
∂ ∂
X2j = ∂x2j − 2x2j−1 ∂t ,
−01 01 .
where ∇x =
|∇Hu|2dxdt ≥ (n − 1)2
and the constant (n - 1)2 in (1.6) is sharp in the sense of
(n − 1)2 =
Corollary 1.2
There holds, for u ∈ C0∞( H),
|∇Hu|2 ≥ (n − 1)2
Theorem 1.1
+ V(r)y(r) = 0
and the constant 1/4 is sharp in the sense of
H |∇Hu|2 − (n − 1)2
Corollary 1.3
There holds, for u ∈ C0∞( H) and D ≥ R,
|∇Hu|2 ≥ (n − 1)2
and the constant 1/4 is sharp in the sense of
H |∇Hu|2 − (n − 1)2
k=1 H
uH2 |Xux|222 −|Dx14| · ·jk=−·1·1· Xk2H |ux||2x2|X12 |Dx| · · · Xj2 |Dx| .
H |x|2 1 D
2 Proof
+ V(r)y(r) = 0
BR
|∂r f |2dx ≥ (n − 1)2
Observe that if f is radial, i.e., f(x) = f(r), then |∇ f| = |∂r f|. By inequality (1.5),
inequality (2.1) holds.
Now let f ∈ C0∞(BR). If we extend f as zero outside BR, we may consider
f ∈ C0∞(R2n). Decomposing f into spherical harmonics we get (see e.g., [9])
where jk(s) are the orthonormal eigenfunctions of the Laplace-Beltrami operator
with responding eigenvalues
f =
ck = k(N + k − 2), k ≥ 0.
|∂rf |2dx =
k=0 BR
(n − 1)2
f 2 f 2
BR |x|2 dx +BR |x|2 V(|x|)dx =
|fk|2dx ≥ (n − 1)2
Therefore, by (2.2) and (2.3),
|∂rf |2dx =
k=0 BR
= (n − 1)2
BR
This completes the proof of lemma 2.1.
Proof of Theorem 1.1
Recall that the horizontal gradient on Hn is the (2n)-dimensional vector given by
−01 10 .
Therefore, for any φ ∈ C0∞(Hn),
Here we use the fact 〈x, Λx〉 = 0 since Λ is a skew symmetric matrix.
Since u ∈ C0∞( H), for every t Î ℝ,u(·, t) ∈ C0∞(BR). By Lemma 2.1,
|∂ru|2dx ≥ (n − 1)2
|∂ru|2dxdt ≥ (n − 1)2
By (2.4) and the pointwise Schwartz inequality, we have
|∂ru| = | x, ∇xu | = | x, ∇Hu |
|x| |x|
Therefore, we obtain, by (2.6)
≤ |∇Hu|.
(n − 1)2
|∇Hu|2dxdt.
H |∇xφ(|x|)|2w2(t) + 4 H |x|2φ2(|x|)(w (t))2
= BR |∇xφ(|x|)|2dx + 4 BR |x|2φ2(|x|)dx
+∞ (w (t))2
we obtain
inf
w(t)∈C0∞(R)\{0} R |w(t)|2dt
= 0,
H |∇Hu|2dxdt
= (n − 1)2.
The proof of Theorem 1.1 is completed.
Proof of Corollary 1.2
H |∇xφ(|x|)|2w2(t) − (n − 1)2
φ2 1 k−1 φ2
BR |∇xφ(|x|)|2 − (n − 1)2 BR |x|2 − 4 j=1 BR |x|2
inf
w(t)∈C0∞(R)\{0} R |w(t)|2dt
= 0,
H |∇Hu|2 − (n − 1)2
ik=1 log(i) R −2
|x|
φ2 1 k−1 φ2
BR |∇xφ(|x|)|2 − (n − 1)2 BR |x|2 − 4 j=1 BR |x|2
Here we use the fact that the sharp constant in inequality (1.3) is 1/4 (see [4]). This
completes the proof of Corollary 1.2.
Proof of Corollary 1.3
we have
H |∇Hu|2 − (n − 1)2
H |∇xφ(|x|)|2w2(t) − (n − 1)2
φ2w2(t) X12 |Dx| · · · Xj2 |Dx|
H |x|2
H |x|2φ2(|x|)(w (t))2
BR |x|2φ2(|x|) +∞ (w (t))2
BR |x|2 X12 |D| · · · Xj2 |Dx| · −+∞∞ w2(t)dt
φ2 x
−∞
H |∇Hu|2 − (n − 1)2
= 0.
This completes the proof of Corollary 1.3.
Competing interests
The author declares that they have no competing interests.
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1. Brezis , H, Vázquez, JL : Blowup solutions of some nonlinear elliptic problems . Rev Mat Univ Comp Madrid . 10 , 443 - 469 ( 1997 )
2. Adimurthi , N, Chaudhuri , N, Ramaswamy , N: An improved Hardy Sobolev inequality and its applications . Proc Amer Math Soc. 130 , 489 - 505 ( 2002 ). doi:10.1090/S0002- 9939 - 01 - 0613 (...truncated)