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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 158281, 17 pages doi:10.1155/2010/158281 Research Article Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups Yongyang Jin1 and Yazhou Han2 1 2 Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310032, China Department of Mathematics, College of Science, China Jiliang University, Hangzhou 310018, China Correspondence should be addressed to Yongyang Jin, [email protected] Received 19 March 2010; Accepted 15 June 2010 Academic Editor: Nikolaos Papageorgiou Copyright q 2010 Y. Jin and Y. Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove a sharp weighted Rellich inequality associated with a class of Greiner-type vector fields on H-type groups. We also obtain some weighted Hardy- and Rellich-type inequalities on nonisotropic Heisenberg groups. As an application, we get a Rellich-Sobolev-type inequality on Heisenberg groups. 1. Introduction The study of partial diﬀerential operators constructed from noncommutative vector fields satisfying the Hörmander condition 1 has had much development. We refer to 2, 3 and the references therein for a systematic account of the study. Recently there have been considerable interests in studying the sub-Laplacians as square sums of vector fields that are not invariant or do not satisfy the Hörmander condition. Among the examples of such sub-Laplacians are the Grushin operators, Greiner-type operators, and the sub-Laplacian constructed by Kohn 4. Those noninvariant sub-Laplacians also appear naturally in complex analysis. In 5 Beals n et al. considered the CR operators {Zj , Zj }j1 on R2n1 as boundary of the complex domain ⎧ ⎪ ⎨ ⎞k ⎫ ⎪ n ⎬ 2⎠ ⎝ zj , > ⎪ ⎭ j1 ⎛ z1 , . . . , zn1 ∈ Cn1 : Im zn1 ⎪ ⎩ 1.1 2 Journal of Inequalities and Applications where Zj 1/2Xj − iYj , Xj ∂ ∂ 2kyj |z|2k−2 , ∂xj ∂t Yj ∂ ∂ − 2kxj |z|2k−2 , ∂yj ∂t 1.2 and k is a positive integer. The space R2n1 has a natural structure of a Heisenberg group, but the vector fields are not left or right invariant. In 6 Zhang and Niu studied the Greiner vector fields on R2n1 for general parameter k ≥ 1 and got the corresponding Hardy-type inequality. Note that for nonintegral k these vector fields do not satisfy the Hörmander condition and are not smooth. H-type groups were introduced by Kaplan 7 as direct generalizations of Heisenberg groups. In 8 we define a class of vector fields X see 2.5 on H-type groups generalizing the vector fields 1.2 considered in 5, 6 and find the fundamental solution of the corresponding p-Laplacian with singularity at the identity element. Also we prove a Hardy-type inequality associated to X. The goal of the present paper is to continue our study on analysis associated with Greiner-type vector fields X introduced in 8. We will throughout study the Rellich inequality which is a generalization of Hardy inequality to higher-order derivatives. They have various applications in the study of elliptic and parabolic PDEs. The classical Rellich inequality in Rn states that for n ≥ 5 and φ ∈ C0∞ Rn \ {0}, Δφx2 dx ≥ Rn nn − 4 4 φx2 2 Rn |x|4 dx. 1.3 The constant n2 n − 42 /16 is sharp and is never achieved. Davies and Hinz 9 generalized 1.3 to the Lp case and showed that for any p ∈ 1, n/2 there holds Δφxp dx ≥ Rn p φxp n p − 1 n − 2p dx. p2 |x|2p Rn 1.4 See also the works in 10–13. In a recent paper 14 Yang obtains an L2 version of Rellich inequality associated with the left-invariant vector fields in the setting of Heisenberg group, there is a similar L2 Rellich inequality on the general carnot group in 15 with diﬀerent approach. A natural question is to find an Lp version of Rellich inequality in this general setting. The main purpose of the present paper is to prove some weighted Lp -Rellich inequalities associated with Greiner-type vector fields on H-type groups. Our approach depends on the fundamental solution of the corresponding square operator and the weighted Hardy inequality proved in our earlier paper 8. We prove also some weighted Lp Hardy and Rellich inequalities on nonisotropic Heisenberg groups by a diﬀerent method caused by the absence of the explicit representation formula for fundamental solution. As an application, we get a Rellich-Sobolevtype inequality on Heisenberg groups. The plan of the paper is as follows. In Section 2 we introduce a class of Greinertype vector field and prove the corresponding weighted Lp -Rellich inequality on H-type groups; Section 3 is devoted to the proof of weighted Hardy- and Rellich-type inequalities Journal of Inequalities and Applications 3 on nonisotropic Heisenberg groups and a Rellich-Sobolev-type inequality on Heisenberg groups. 2. Rellich Inequality on H-Type Groups We recall that a simply connected nilpotent group G is of Heisenberg type, or of H-type, if its Lie algebra n V ⊕ t is of step two, V, V ⊂ t, t, V 0 and if there is an inner product · , · on n such that the linear map J : t −→ EndV , 2.1 Jt u, v t, u, v u, v ∈ V, z ∈ t, 2.2 Jt2 −|t|2 Id 2.3 defined by the condition satisfies for all t ∈ t, where |t|2 t, t. Groups of H-type were introduced by Kaplan in 7 as direct generalizations of Heisenberg groups, and they have been studied quite extensively; see 16–19 and the references therein. We identify G with its Lie algebra n via the exponential map, exp : V ⊕ t → G. The Lie group product is given by u, tv, s 1 u v, t s u, v . 2 2.4 For g ∈ G, we write g zg, tg ∈ V ⊕ t. In 8 the authors constructed a family of Greiner-type vector fields X {X1 , . . . , Xm } on G: 1 Xj ∂j k|z|2k−2 ∂z,ej , 2 j 1, 2 . . . , m, 2.5 where ∂j ∂ej , ∂z,ej are the directional derivatives, {ej }j1,...,m is an orthonormal basis of V , and k ≥ 1 is a fixed parameter. If k 1, Xj are the left-invariant vector fields defined by the orthonormal basis {ej }m j1 on V. The corresponding degenerate p-sub-Laplacian is Lp,k u divX |∇X u|p−2 ∇X u , 2.6 4 Journal of Inequalities and Applications where ∇X u X1 u, . . . , Xm u, divX u1 , . . . , um m Xj uj . 2.7 j1 We denote ΔX L2,k m Xi2 . 2.8 i1 There is a one-parameter group of dilations {δλ : λ > 0} on G: δλ : z, t −→ w, s λz, λ2k t . 2.9 The volume element is transformed by δ via 2.10 dw ds λQ dz dt, where Q : m2kq with m dim V and q dim t. Q will be called the degree of homogeneity and is the homogeneous dimension if k 1. We define a corresponding homogeneous norm by 1/4k d dz, t : |z|4k 16|t|2 . 2.11 Remark 2.1. Note that when p 2 and k 1, Lp,k becomes the sub-Laplacian ΔG on the Htype group G. If p 2, q 1, and k 2, 3, . . ., Lp,k is a Greiner operator see 5, 20. Also we note that vector fields {Xj } are neither left nor right invariant and they do not satisfy Hörmander’s condition for k > 1, k / ∈ Z. The main results in 8 are the following. Theorem 2.2. Let G be an H-type group, k ≥ 1, and Q m 2kq. Then for 1 < p < ∞, Γp ⎧ ⎪ ⎨Cp dp−Q/p−1 , p/ Q, ⎪ ⎩CQ log 1 , d pQ 2.12 is a fundamental solution of Lp,k with singularity at the identity element 0 ∈ G. Here dz, t is defined in 2.11, −1/Q−1 p − 1 −1/p−1 , CQ − σQ , σp p−Q q−1/2 qm/2 π Γ 2k − 1p m /4k 1 σp . 4 Γm/2Γ 2k − 1p Q /4k Cp 2.13 Journal of Inequalities and Applications 5 Theorem 2.3. Let G be an H-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p < Q α. Then the following Hardy-type inequality holds for Φ ∈ C0∞ G \ {0}: p d |∇X Φ| ≥ α G Qα−p p p G d α |z| d 2k−1p p Φ , d 2.14 where ∇X f X1 f, X2 f, . . . , Xm f is the gradient defined by the vector fields 2.5. Moreover, the constant Q α − p/pp is sharp. Based on the above two theorems, we will prove the following Lp version of weighted Rellich inequality on H-type groups. Theorem 2.4. Let G be an H-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p < ∞, 2 − Q < α < min{p − 1Q − 2, Q − 2}. Then the following Rellich-type inequality holds for Φ ∈ C0∞ G \ {0}: G d α4kp−1 |z| p ΔX φ ≥ 2−4kp−1 p p Q α − 2 p − 1 Q − 2 − α dα−4k |z|4k−2 φ . p2 G 2.15 Moreover, the constant Q α − 2p − 1Q − 2 − α/p2 p is sharp. Remark 2.5. In the abelian case G Rn , the above result recovers the classical Rellich inequality 1.4 with dx |x|, α 2 − 2p under the condition 2p < n. Remark 2.6. When we take p 2 and k q 1, our inequality 2.15 is just inequality 1.5 in 14 and inequality 5.2 in 15 for Heisenberg groups. Now we prove Theorem 2.4. Proof. We denote u d2−Q , where d is as in 2.11, then ΔX dα ΔX d2−Q α/2−Q ΔX uα/2−Q α α uα/2−Q−1 ΔX u 2−Q 2−Q α − 1 uα/2−Q−2 |∇X u|2 , 2−Q 2.16 for α > 2 − Q; we have G p p α α α uα/2−Q−1 ΔX uφ − 1 uα/2−Q−2 |∇X u|2 φ G 2−Q G 2−Q 2−Q p α α − 1 uα/2−Q−2 |∇X u|2 φ G 2−Q 2−Q p αα − 2 Q dα−2 |∇X d|2 φ p φ ΔX dα G 2.17 6 Journal of Inequalities and Applications since C2 · u is the fundamental solution of ΔX at the origin by Theorem 2.2; however the left-hand side is G p φ ΔX dα − p G G p−2 pφ φ∇X φ∇X dα d α 2 p−2 p−2 p − 1 φ ∇X φ φ φΔX φ . 2.18 Thus, by 2.17, 2.18, and the corresponding weighted Hardy inequality Theorem 2.3, we have αα−2Q G p dα−2 |∇X d|2 φ −p G p−2 dα φ φΔX φ 2 2 p−2 4p p−1 p/2 α d ∇X φ φ d ∇X φ p2 G G 2 p 4 p−1 Qα−2 2 p−1 2 α p |∇X d| ≥ d φ dα−2 |∇X d|2 φ , Qα−2 2 p 2 p d G G 2.19 p p−1 α this implies that −p G p p−1 2 dα−2 |∇X d|2 φ Q α − 2 − αα − 2 Q p G p p−1 Q α − 2 dα−2 |∇X d|2 φ Q α − 2 − α p G p Q α − 2 p − 1 Q − 2 − α dα−2 |∇X d|2 φ . p G p−2 d φ φΔX φ ≥ α 2.20 Applying Hölder’s inequality, we get p Q α − 2 p − 1 Q − 2 − α dα−2 |∇X d|2 φ 2 p G 1/p 21−p p p−1/p 2 p α−2 α ∇X d ΔX φ ≤ d |∇X d| φ d . d G G 2.21 Noticing that |∇X d| we have thus proved 2.15. |z| d 2k−1 , 2.22 Journal of Inequalities and Applications 7 It remains to show the sharpness of the constant Q α − p /pp . Let B be any constant satisfying the inequality G d α−22p p ΔX φ ≥ B 2−2p |∇X d| G p dα−2 |∇X d|2 φ . 2.23 p We will prove that B ≤ Q α − 2p − 1Q − 2 − α/p2 . The idea is to find functions {uj }∞ j1 so that the diﬀerence between the left- and right-hand sides approximates to 0. Given any positive integer j it is elementary that there exists ψj in C0∞ 0, ∞ such that supp ψj 2−j−1 , 2, ψj x 1 on 2−j , 1, and |ψj x| ≤ C2j , |ψj x| ≤ C22j on 2−j−1 , 2−j , where C is a constant independent of j. Let uj z, t dz, t2−Q−α/p−1/j ψj dz, t. 2.24 Clearly uj ∈ C∞ G \ {0} which is radial. Denoting Cj by 2 − Q − α/p − 1/j, it is easy to see that ∇X uj ⎧ ⎨0, 0 ≤ d < 2−j−1 , or d > 2 ⎩C dCj −1 ∇ d, 2−j < d < 1, j X ⎧ ⎨0, ΔX uj ⎩C C Q − 2dCj −2 |∇ d|2 , j j X 0 ≤ d < 2−j−1 , or d > 2 2.25 2−j < d < 1. Here we used the fact that 2.26 dΔX d Q − 1|∇X d|2 . The left-hand side of the above inequality 2.23 is LHS G 2−j <d<1 2−j−1 <d≤2−j 1≤d<2 2−j <d<1 I II. 2.27 The first integration is 2−j <d<1 p dα−22p |∇X d|2−2p ΔX φ Qα−2 1 p j p p p − 1 Q − 2 − α 1 − d−Q−p/j |∇X d|2 , p j 2−j <d<1 2.28 which can be evaluated as the last computations in the proof of Theorem 1 in 8, and is Qα−2 1 p j p p p − 1 Q − 2 − α 1 − C0 2p − 1j, p j 2.29 8 Journal of Inequalities and Applications where C0 is a positive constant. Similarly, RHS B 2−j <d<1 III IV. 2.30 The first integration is precisely the same as above and is B 2−j <d<1 BC0 2p − 1j. 2.31 It is easy to estimate the error terms which they are all bounded: I, II, III, IV ≤ C. 2.32 The inequality 2.23 now becomes Qα−2 1 p j p p p − 1 Q − 2 − α 1 − C0 2p − 1j I II p j 2.33 ≥ BC0 2p − 1j III IV. Dividing both sides by j and letting j → ∞ prove our claim. The following is an immediate consequence of Theorem 2.4, which is an extension of the uncertainty principle inequality. Corollary 2.7. Let G be a Heisenberg-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p < ∞, 1/p 1/q 1, 2 − Q < α < min{p − 1Q − 2, Q − 2}. Then for all φ ∈ C0∞ G \ {0} the following inequality holds: G p dα2p−1 |∇X d|21−p ΔX φ 1/p G q d2−αq/p |∇X d|2q/p φ 2 Q α − 2 p − 1 Q − 2 − α φ . ≥ 2 p G 1/q 2.34 Remark 2.8. We mention that when p q 2 and k 1, our inequality 2.34 goes back to inequality 5.7 in 15 in the setting of H-type group. We end this section with the following Rellich-type inequality on the polarizable group which can be proved by the same method if we noted Theorem 2.15 and Proposition 2.18 in 21 and the weighted Hardy inequality Theorem 4.1 in 22. Journal of Inequalities and Applications 9 Theorem 2.9. Let G be a polarizable group with homogeneous dimension Q > 4. Suppose that 1 < p < ∞, 2 − Q < α < min{p − 1Q − 2, Q − 2}. Then the following inequality holds for all φ ∈ C0∞ G \ {0}: p 21−p 2 p Q α − 2 p − 1 Q − 2 − α α ∇G N p ΔG φ ≥ φ . N N 2 N N p G G 2.35 α ∇G N Here N u1/2−Q is the homogeneous norm associated with the fundamental solution u for the Kohn p sub-Laplacian. Moreover, the constant Q α − 2p − 1Q − 2 − α/p2 is sharp. 3. Rellich Inequality on Nonisotropic Heisenberg Groups Let a a1 , a2 , . . . , an , a1 , a2 , . . . , an > 0. Let H Ha R2n ⊕ R be the corresponding nonisotropic Heisenberg group with the product ⎞ ⎛ n ζ, t ◦ η, s ⎝ζ η, t s 2 aj ζjn ηj − ζj ηjn ⎠. 3.1 j1 We consider the following nonisotropic Greiner-type vector fields: ⎛ ⎞k−1 n 2 ∂ ∂ , Xj 2kaj yj ⎝ aj zj ⎠ ∂xj ∂t j1 ⎛ Yj ⎞k−1 n 2 ∂ − 2kaj xj ⎝ aj zj ⎠ ∂yj j1 j 1, . . . , n, 3.2 ∂ , ∂t where x, y, t ∈ R2n ⊕ R, x x1 , x2 , . . . , xn , y y1 , y2 , . . . , yn . Denote ∇H u X1 u, . . . , Xn u, Y1 u, . . . , Yn u, divH u1 , . . . , u2n n Xj uj Yj unj , j1 ⎞1/4k ⎞2k n 2 ⎟ ⎜ , d ⎝⎝ aj zj ⎠ t2 ⎠ ⎛⎛ 3.3 j1 n Q 2 aj 2k. j1 For further information on the nonisotropic Heisenberg group, see, for example, 23, 24. 10 Journal of Inequalities and Applications In this section we prove firstly the Lp Hardy-type inequality associated with the Greiner-type vector fields 3.2 on the nonisotropic Heisenberg group H. Theorem 3.1. Let H Ha be the anisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Let α ∈ R, 2 ≤ p < Q α and Φ ∈ C0∞ H \ {0}. Then the following inequality is valid: dα |∇L Φ|p ≥ H where d n j1 ⎛ p Qα−p p H 1/4k aj |zj |2 2k t2 ⎞p/2 ⎛ ⎞k−1p n n 2 2 dα−2kp ⎝ a2j zj ⎠ ⎝ aj zj ⎠ |Φ|p , j1 , Q2 n j1 3.4 j1 aj 2k. For the proof of the above inequality, we need the following lemma which can be proved by a similar method in 25. Lemma 3.2. Let w ≥ 0 be a weight function in Ω ⊂ G and Lp,k,w u divH |∇H u|p−2 w∇H u. Suppose that for some λ > 0, there exists v ∈ C∞ Ω, v > 0 such that −Lp,k,w v ≥ λgvp−1 3.5 for some g ≥ 0, in the sense of distribution acting on nonnegative test functions. Then for any u ∈ 1,p HW0 Ω, w, it holds that Ω |∇H u|p w ≥ λ 1,p Ω g|u|p , where HW0 Ω, w denote the closure of C0∞ Ω in the norm 3.6 1/p Ω |∇H u|p w . We now prove Theorem 3.1. Proof. Take w dα and v dp−Q−α/p . Noting that LH,p,w v ! Xi |∇L v|p−2 wXi v Yi |∇L v|p−2 wYi v , n i1 " Xi d Yi d 2 2k−1 2 k−1 n aj zj ai xi ai yi t j1 aj zj # d4k−1 # " 2 2k−1 2 k−1 n n ai yi − ai xi t j1 aj zj j1 aj zj |∇L d|2 n j1 n j1 2 n a2j zj j1 d4k−2 d4k−1 2 2k−2 aj zj , 3.7 , , Journal of Inequalities and Applications 11 we get α − p −Qα/p p−2 p−2 p − Q − α −Qα/p Xi d d Xi d −LH,p,w v − |∇L d| p p i1 p−2 p−2 p − Q − α α Q α − p −Qα/p −Qα/p Yi d d d Yi d |∇L d| p p n d Qα−p p α Q 3.8 p−1 {I1 I2 I3 I4 }, where I1 Q α − pQ 2 − p 2k − 1 dQα−pQ−p/p2−p2k−1 p ⎛ ⎞p−2/2 ⎛ ⎞p−2k−1 n n 2 ⎝ aj zj 2 ⎠ × ⎝ a2 zj ⎠ |∇L d|2 ; j j1 j1 ⎛ ⎞p−4/2 ⎛ ⎞p−2k−1 n n Qα−pQ/p2−p2k−1 2 2 ⎝ a2 zj ⎠ ⎝ aj zj ⎠ I2 p − 2 d j j1 × j1 n a2j xj · Xj d yj · Yj d ; j1 ⎛ ⎞p−2/2 ⎛ ⎞p−2k−1−1 n n 2 2 Qα−pQ/p2−p2k−1 2 ⎝ a zj ⎠ ⎝ aj zj ⎠ I3 2k − 1 p − 2 d j j1 × j1 n aj xj · Xj d yj · Yj d ; j1 ⎛ ⎞p−2/2 ⎛ ⎞p−2k−1 n n n 2 ⎝ aj zj 2 ⎠ Xj2 d Yj2 d . I4 dQα−pQ/p2−p2k−1 ⎝ a2j zj ⎠ j1 j1 j1 3.9 By direct computations we have n aj xj · Xj d yj · Yj d n j1 d4k−1 j1 n a2j xj · Xj d yj · Yj d j1 2 2k−1 n 2 2 aj zj j1 aj zj n j1 2 2k−1 n 3 2 aj zj j1 aj zj d4k−1 , , 12 Journal of Inequalities and Applications n j1 ⎞2k−1 n ⎛ 2 n 2 2 n n 2 j1 aj j1 aj zj j1 aj zj 2 2 2 ⎠ ⎝ Xj d Yj d aj zj . d4k−1 j1 3.10 Then Qα−p −LH,p,w v p p−1 ⎛ ⎞p−4/2 ⎛ ⎞k−1p n n 2 ⎝ aj zj 2 ⎠ dQα/p−1−Q2k−12−p ⎝ a2 zj ⎠ · II, j j1 j1 3.11 where II ⎧ ⎪ ⎨ Q α ⎪ ⎩ p − Q 2k − 1 2 − p ⎛ ⎛ ⎞2 ⎞⎛ ⎞ n n n 2 2 2 2 3 ⎝ a zj ⎠ p − 2 ⎝ aj zj ⎠⎝ a zj ⎠ j j j1 j1 j1 ⎛ ⎞2 ⎛ ⎞ n n 2 2 2k − 1 p − 2 ⎝ a2j zj ⎠ ⎝ a2j zj ⎠ j1 j1 ⎞⎛ ⎞⎫⎫ n n n ⎬ 2 2 ⎬⎪ 2 ⎠ ⎝ ⎝ ⎠ ⎝ . 2 aj × 2k − 1 aj zj aj zj ⎠ ⎭⎪ ⎩ ⎭ j1 j1 j1 ⎧ ⎨ ⎛ ⎞ ⎛ 3.12 Using Cauchy inequality ⎛ ⎞⎛ ⎞ ⎛ ⎞2 n n n 2 2 2 3 2 ⎝ aj zj ⎠⎝ a zj ⎠ ≥ ⎝ a zj ⎠ j j j1 j1 3.13 j1 and that since, by assumption, aj ≤ 1, j 1, . . . , n n n 2 2 aj zj ≥ a2j zj , j1 3.14 j1 we find ⎞2 ⎛ n Q α − p ⎝ 2 II ≥ a2j zj ⎠ , p j1 3.15 Journal of Inequalities and Applications 13 so for p ≥ 2, −LH,p,w v ≥ Qα−p p Qα−p p p dp−Q−α/p p−1 ⎛ ⎞p/2 ⎛ ⎞k−1p n n 2 2 dα−2kp ⎝ a2j zj ⎠ ⎝ aj zj ⎠ j1 p j1 3.16 vp−1 dα−p |∇L d|p . Hence Theorem 3.1 follows from Lemma 3.2 with λ Q α − p/pp , g dα−p |∇L d|p . Now it is time to prove the following Rellich inequality on nonisotropic Heisenberg group. Theorem 3.3. Let H Ha be a nonisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Suppose that α ∈ R, 2 − Q < α ≤ 0 and u ∈ C0∞ H \ {0}. Then the following inequality is valid: H dα−4k4kp p 2k−1p−1 |ΔH u| 2 n 2 j1 aj zj ⎛ ⎞2k−1 p n p − 1 Q − 2 − α Q α − 2 2 dα−4k ⎝ a2j zj ⎠ |u|p , p2 H j1 ≥ where d n aj |zj |2 j1 2k 1/4k t2 , Q2 n j1 3.17 aj 2k. Remark 3.4. If p 2, α −2, and k ai 1 i 1, . . . , n, then we get the Rellich inequality on the Heisenberg group Hn with homogeneous dimension Q 2n 2: d2 Hn |z| 2 |ΔHn u| ≥ 2 QQ − 4 4 2 Hn |z|2 2 |u| . d6 3.18 We also mention that to our knowledge, even in the special case k aj 1, j 1, 2, . . . , n, our inequality 3.17 is new. We now give the proof of Theorem 3.3. Proof. We have p |u|p α − 1dα−2 |∇H d|2 dα−1 ΔH d |u| ΔX d α H α H 3.19 p α−2 αα − 1 |u| d H 2 p α−1 |∇H d| α |u| d H ΔH d. 14 Journal of Inequalities and Applications By a direct computation we have " Xi d n j1 2 2k−1 2 k−1 n aj zj ai xi ai yi t j1 aj zj # , d4k−1 " Yi d n j1 2 2k−1 2 k−1 n zj aj zj ai yi − a ai xi t j j1 , d4k−1 n j1 |∇H d|2 2 n 2 2k−2 a2j zj j1 aj zj d4k−2 # , ⎞2k−1 n 2 n 2k − 1 n a2 zj 2 n 2 j1 aj 2 j1 aj zj j1 j ΔH d ⎝ aj zj ⎠ . d4k−1 j1 ⎛ 3.20 Noting that ai ≤ 1 then dΔH d ≥ Q − 1|∇H d|2 ; 3.21 we take α < 0 and thus deduce from 3.19 that |u|p dα−2 |∇H d|2 . |u|p ΔX dα ≤ αQ α − 2 H 3.22 H On the other hand, |u|p ΔH dα −p H |u|p−2 u∇H u∇H dα H p d α p − 1 |u|p−2 |∇H u|2 |u|p−2 uΔH u . 3.23 H Combining with 3.22, 3.23, and the L2 weighted Hardy inequality associated with the Greiner-type vector fields 3.2 on the nonisotropic Heisenberg group 3.4, then we only need to do the same steps as in Theorem 2.4 and thus finish the proof of Theorem 3.3. Journal of Inequalities and Applications 15 The following is also an uncertainty principle type inequality which is an immediate consequence of Theorem 3.3. Corollary 3.5. Let H Ha be an anisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Then for u ∈ C0∞ H \ {0}, 1/p 1/q 1 1 < p < ∞, the following inequality holds: ⎛ ⎛ ⎜ ⎜ ⎝ ⎝ n H ⎞p−1 d4k ⎟ 2k−1 ⎠ 2 2 j1 aj zj ⎞1/p ⎛ ⎛ ⎟ ⎜ ⎜ |ΔH u|p ⎠ ⎜ ⎝ ⎝ n H ⎞1/p−1 ⎞1/q ⎟ |u|q ⎟ ⎠ d4k ⎟ 2k−1 ⎠ 2 2 j1 aj zj ≥ p − 1 Q − 22 |u|2 , p2 H where d n 1/4k 2 j1 3.24 aj |zj |2k t2 , Q2 n j1 aj 2k. By a similar method, we can get the following inequality which does not contain the weight |∇H d|, so we omit the proof. Theorem 3.6. Let H Ha be a nonisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Let α ∈ R, 2 − Q < α ≤ 0 and u ∈ C0∞ H \ {0}. Then the following inequality is valid: p p − 1 2 ni1 ai − α 2 ni1 ai α − 2p |z|α−2p |u|p . p2 H α p |z| |ΔH u| ≥ H 3.25 With the help of Theorem 3.6, we can also obtain a Rellich-Sobolev-type inequality on the Heisenberg group. Corollary 3.7. Let H Hn be the Heisenberg group with homogeneous dimension Q 2n 2. Suppose that 1 < p < Q − 2/2, 0 ≤ s ≤ p, ps pQ − 2s/Q − 2p. Then there exists a positive constant C such that for any u ∈ C0∞ H \ {0}, the following inequality holds: |u|ps H 1/ps |ΔH u|p ≤C |z|2s 1/p 3.26 . H Proof. By Hölder inequality we have |u|ps H |z|2s |u|s H |z| Qp−s/Q−2p |u| 2s |u|p ≤ H |z|2p s/p |u|pQ/Q−2p H p−s/p . 3.27 16 Journal of Inequalities and Applications Thanks to the Rellich inequality 3.25 with α 0, ai 1 i 1, 2, . . . , n and the Sobolev inequality on the nilpotent Lie group Chapter IV Theorem 3.3.1 in 26 we get |u|ps H |z|2s |H u|p ≤C s/p H H |H u|p Qp−s/pQ−2p |H u|p C Q−2s/Q−2p . H 3.28 Thus we obtain the desired result. 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