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volume 4, issue 5, article 103, 2003.

Received 21 January, 2003;

accepted 10 June, 2003.

Communicated by:L. Pick

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Journal of Inequalities in Pure and Applied Mathematics

SOME HARDY TYPE INEQUALITIES IN THE HEISENBERG GROUP

YAZHOU HAN AND PENGCHENG NIU

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072,

P.R. China.

EMail:taoer558@163.com

EMail:pengchengniu@yahoo.com.cn

2000c Victoria University ISSN (electronic): 1443-5756 009-03

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Some Hardy Type Inequalities in the Heisenberg Group Yazhou Han and Pengcheng Niu

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J. Ineq. Pure and Appl. Math. 4(5) Art. 103, 2003

http://jipam.vu.edu.au

Abstract

Some Hardy type inequalities on the domain in the Heisenberg group are established by using the Picone type identity and constructing suitable auxiliary functions.

2000 Mathematics Subject Classification:35H20.

Key words: Hardy inequality, Picone identity, Heisenberg group.

The research supported in part by National Nature Science Foundation and Natural Science Foundation of Shaanxi Province, P.R. China.

1. Introduction

The Hardy inequality in the Euclidean space (see [3], [4], [7]) has been established using many methods. In [1], Allegretto and Huang found a Picone’s identity for thep-Laplacian and pointed out that one can prove the Hardy inequality via the identity. Niu, Zhang and Wang in [6] obtained a Picone type identity for thep-sub-Laplacian in the Heisenberg group and then established a Hardy type inequality. Whenp= 2, the result of [6] coincides with the inequality in [2]. As stated in [1], the Picone type identity allows us to avoid postulating regularity conditions on the boundary of the domain under consideration. Since there is a presence of characteristic points in the sub-Laplacian Dirichlet problem in the Heisenberg group (see [2]), we understand that such an identity is especially useful.

We recall that the Heisenberg groupH_nof real dimensionN = 2n+ 1, n∈ N, is the nilpotent Lie group of step two whose underlying manifold isR²ⁿ⁺¹. A basis for the Lie algebra of left invariant vector fields onH_nis given by

X_j = ∂

∂xj

+ 2y_j ∂

∂t, Y_j = ∂

∂yj

−2x_j ∂

∂t, j = 1,2, . . . , n.

The number Q= 2n+ 2is the homogeneous dimension ofH_n. There exists a Heisenberg distance

d((z, t),(z⁰, t⁰)) = n

(x−x⁰)²+ (y−y⁰)²2

+ [t−t⁰−2(x·y⁰ −x⁰ ·y)]² o¹₄

between(z, t)and(z⁰, t⁰). We denote the Heisenberg gradient by

∇Hn = (X1, . . . , Xn, Y1, . . . , Yn).

In this note we give some Hardy type inequalities on the domain in the Heisenberg group by considering different auxiliary functions.

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2. Hardy Inequalities

First we state two lemmas given in [6] which will be needed in the sequel.

Lemma 2.1. Let Ω be a domain inHn, v > 0, u ≥ 0 be differentiable inΩ.

Then

(2.1) L(u, v) =R(u, v)≥0,

where

L(u, v) =|∇_H_nu|^p+ (p−1)u^p

v^p|∇_H_nv|^p−pu^p−1

v^p−2∇_H_n· |∇_H_nv|^p−2∇_H_nv, R(u, v) =|∇_H_nu|^p− ∇_H_n

u^p v^p−1

· |∇_H_nv|^p−2∇_H_nv.

Denote thep-sub-Laplacian by∆_H_n_,pv =∇_H_n ·(|∇_H_nv|^p−2∇_H_nv).

Lemma 2.2. Assume that the differentiable function v > 0satisfies the condi- tion−∆_H_n_,pv ≥λgv^p−1, for someλ >0and nonnegative functiong. Then for everyu∈C₀^∞(Ω), u≥0,

(2.2)

Z

Ω

|∇_H_nu|^p ≥λ Z

Ω

g|u|^p.

LetB_R ={(z, t)∈H_n|d((z, t),(0,0)) < R}be the Heisenberg group and δ(z, t) = dist ((z, t), ∂B_R), (z, t) ∈ B_R, in the sense of distance functions on the Heisenberg group.

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Theorem 2.3. LetΩ =B_R\{(0,0},p > 1. Then for everyu∈C₀^∞(Ω), (2.3)

Z

Ω

|∇_H_nu|^p ≥

p−1 p

pZ

Ω

|z|^p d^p

|u|^p δ^p , where|z|=p

x² +y², d=d((z, t),(0,0)).

Proof. We first consideru≥0. The following equations are evident:

(2.4)











X_jd=d⁻³(|z|²x_j +y_jt), Y_jd =d⁻³(|z|²y_j−x_jt), X_j²d=−3d⁻⁷(|z|²x_j +y_jt)² +d⁻³ |z|² + 2x²_j + 2y_j²

, Y_j²d=−3d⁻⁷(|z|²y_j−x_jt)²+d⁻³ |z|²+ 2x²_j + 2y_j²

, j = 1, . . . , n and

(2.5) |∇_H_nd|=|z|d⁻¹, ∆_H_nd= (Q−1)d⁻³|z|². Choosev(z, t) =δ(z, t)^β = (R−d)^β, in whichβ = ^p−1_p , one has

X_jv =−βδ^β−1X_jd, Y_jv =−βδ^β−1Y_jd, j = 1, . . . , n,

∇_H_nv =−βδ^β−1∇_H_nd, |∇_H_nv|=|β|δ^β−1|z|d⁻¹, and

−∆_H_nv =−∇_H_n· |∇_H_nv|^p−2∇_H_nv

=−∇Hn· −β|β|^p−2δ^{(β−1)(p−1)}|z|^p−2d^2−p∇Hnd

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=β|β|^p−2

−(β−1)(p−1)δ(β−1)(p−1)−1|z|^p−2d^2−p|∇_H_nd|² +δ^{(β−1)(p−1)}d^2−p∇_H_n |z|^p−2

· ∇_H_nd + (2−p)δ^{(β−1)(p−1)}|z|^p−2d^1−p|∇_H_nd|² +δ^{(β−1)(p−1)}|z|^p−2d^2−p4_H_nd

.

From the fact ∇_H_n(|z|^p−2)· ∇_H_nd = (p−2)|z|^p−4d⁻³|z|⁴ = (p−2)|z|^pd⁻³ and (2.5), it follows that

−∆_H_nv =β|β|^p−2

−(β−1)(p−1)δ(β−1)(p−1)−1|z|^pd^−p + (p−2)δ^{(β−1)(p−1)}|z|^pd^−1−p

−(p−2)δ^{(β−1)(p−1)}|z|^pd^−1−p + (Q−1)δ^{(β−1)(p−1)}|z|^pd^−1−p

=β|β|^p−2

−(β−1)(p−1) + (Q−1)δ d

|z|^p d^p

v^p−1 δ^p

=

p−1 p

p−1 p−1

p + (Q−1)δ d

|z|^p d^p

v^p−1 δ^p

≥

p−1 p

p

|z|^p d^p

v^p−1 δ^p .

The desired inequality (2.3) is obtained by Lemma2.2. For generalu, by letting u=u⁺−u⁻, we directly obtain (2.3).

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Theorem 2.4. LetΩ = H_n\{B_H_n_,R}, Q > p >1. Then for everyu∈C₀^∞(Ω), there exists a constantC >0, such that

(2.6)

Z

Ω

|∇_H_nu|^p ≥C Z

Ω

|z|^p d^p

|u|^p d^2p. Proof. Suppose thatu ≥ 0. Takev = log _R^dα

, R < d = d((z, t),(0,0)) <

+∞, α <0. Using (2.4) and (2.5) show that

∇_H_nv = R

d α

α d

R ^α−1

1

R∇_H_nd= α d∇_H_nd,

|∇_H_nv|=|α||z|d⁻²,

−4_H_nv =−∇_H_n· |∇_H_nv|^p−2∇_H_nv

=−α|α|^p−2∇_H_n · |z|^p−2d^{−2(p−2)−1}∇_H_nd

=−α|α|^p−2

(p−2)|z|^p−3d^2(2−p)−1∇_H_n(|z|)· ∇_H_nd + (2(2−p)−1)|z|^p−2d^2(1−p)|∇_H_nd|² +|z|^p−2d^2(2−p)−14Hnd

.

Since∇_H_n(|z|)· ∇_H_nd=|z|³d⁻³, the last equation above becomes

−4_H_nv =−α|α|^p−2

(p−2)|z|^p−3d^2(2−p)−1|z|³d⁻³ + (3−2p)|z|^p−2d^2(1−p)|z|²d⁻² + (Q−1)|z|^p−2d^3−2p|z|²d⁻³

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=−α|α|^p−2|z|^pd^−2p(p−2 + 3−2p+Q−1)

=−α|α|^p−2(Q−p)|z|^pd^−2p. (2.7)

Noting

d→+∞lim v^p−1

d^p = 0,

there exists a positive numberM ≥ R, such that ^v^p−1_dp < 1, ford > M. Since

v^p−1

d^p is continuous on the interval [R, M], we find a constantC⁰ >0, such that

v^p−1

d^p < C⁰. Pick outC⁰⁰ = max{C⁰,1}and one hasv^p−1 < C⁰⁰d^p in Ω. This leads to the following

−∆_H_nv ≥C|z|^p d^2p

v^p−1 d^p ,

whereC = ^−α|α|^p−2_C00^(Q−p), and to (2.6) by Lemma2.2. A similar treatment for generalucompletes the proof.

In particular,α = p−Q (1< p < Q)satisfies the assumption in the proof above.

Theorem 2.5. LetΩbe as defined in Theorem2.4andp≥Q. Then there exists a constantC > 0, such that for everyu∈C₀^∞(Ω),

(2.8)

Z

Ω

|∇_H_nu|^p ≥C Z

Ω

|z|^p d^p log _R^dp

|u|^p d^p .

Proof. It is sufficient to show that (2.8) holds foru ≥ 0. Choosev =φ^α, φ = log _R^d, whereR < d <+∞, 0< α <1. We know that from (2.4) and (2.5),

∇_H_nφ =d⁻¹∇_H_nd, |∇_H_nφ|=d⁻²|z|,

∆_H_nφ =d⁻¹4_H_nd−d⁻²|∇_H_nd|² = (Q−2)|z|²d⁻⁴.

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This allows us to obtain

−∆_H_nv =−∇_H_n· |∇_H_nv|^p−2∇_H_nv

=−∇_H_n· |αφ^α−1∇_H_nφ|^p−2αφ^α−1∇_H_nφ

=−α|α|^p−2∇_H_n· φ^{(α−1)(p−1)}|z|^p−2d^2(2−p)∇_H_nφ

=−α|α|^p−2

(α−1)(p−1)φ(α−1)(p−1)−1|z|^p−2d^2(2−p)|∇Hnφ|² + (p−2)φ^{(α−1)(p−1)}|z|^p−3d^2(2−p)∇_H_n(|z|)· ∇_H_nφ + 2(2−p)φ^{(α−1)(p−1)}|z|^p−2d^2(2−p)−1∇_H_nd· ∇_H_nφ +φ^{(α−1)(p−1)}|z|^p−2d^2(2−p)4_H_nφ

=−α|α|^p−2

(α−1)(p−1)φ(α−1)(p−1)−1|z|^p−2d^2(2−p)|z|²d⁻⁴ + (p−2)φ^{(α−1)(p−1)}|z|^p−3d^2(2−p)|z|³d⁻⁴ + 2(2−p)φ^{(α−1)(p−1)}|z|^p−2d^2(2−p)−1|z|²d⁻³ +φ^{(α−1)(p−1)}|z|^p−2d^2(2−p)(Q−2)|z|²d⁻⁴

=−α|α|^p−2v^p−1 φ^p

|z|^p

d^2p {(α−1)(p−1) + (p−2)φ +2(2−p)φ+ (Q−2)φ}

=−α|α|^p−2v^p−1 φ^p

|z|^p

d^2p {(α−1)(p−1) + (Q−p)φ},

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Taking into account that 0< α <1andp≥Q, we have

−α|α|^p−2(Q−p)φ≥0, and therefore

−4_H_nv ≥ −α|α|^p−2(α−1)(p−1)v^p−1 φ^p

|z|^p

d^2p =Cv^p−1 φ^p

|z|^p d^2p,

whereC =−α|α|^p−2(α−1)(p−1). An application of Lemma2.2completes the proof of Theorem2.5.

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References

[1] W. ALLEGRETTO AND Y.X. HUANG, A Picone’s identity for the p- Laplacian and applications, Nonlinear Anal., 32 (1998), 819–830.

[2] N. GAFOFALO AND E. LANCONELLI, Frequency functions on the Heisenberg group, the uncertainty priciple and unique continuation, Ann.

Inst. Fourier (Grenoble), 40 (1990), 313–356.

[3] G.H. HARDY, Note on a Theorem of Hilbert, Math. Zeit., 6 (1920), 314–

317.

[4] G.H. HARDY, An inequality betweeen integrals, Messenger of Mathemat- ics, 54 (1925), 150–156.

[5] D. JERISON, The Dirichlet problem for the Laplacian on the Heisenberg group, I, II, Journal of Functional Analysis, 43 (1981), 97–141; 43 (1981), 224–257.

[6] P. NIU, H. ZHANGANDY. WANG, Hardy type and Rellich type inequali- ties on the Heisenberg group, Proc. A.M.S., 129 (2001), 3623–3630.

[7] A. WANNEBO, Hardy inequalities, Proc. A.M.S., 109 (1990), 85–95.