Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups
© Yongyang Jin and Yazhou Han. 2010
Received: 19 March 2010
Accepted: 15 June 2010
Published: 30 June 2010
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.
and is a positive integer. The space has a natural structure of a Heisenberg group, but the vector fields are not left or right invariant. In  Zhang and Niu studied the Greiner vector fields on for general parameter and got the corresponding Hardy-type inequality. Note that for nonintegral these vector fields do not satisfy the Hörmander condition and are not smooth.
H-type groups were introduced by Kaplan  as direct generalizations of Heisenberg groups. In  we define a class of vector fields (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 -Laplacian with singularity at the identity element. Also we prove a Hardy-type inequality associated to .
In a recent paper  Yang obtains an version of Rellich inequality associated with the left-invariant vector fields in the setting of Heisenberg group, there is a similar Rellich inequality on the general carnot group in  with different approach. A natural question is to find an version of Rellich inequality in this general setting. The main purpose of the present paper is to prove some weighted -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 . We prove also some weighted 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-Sobolev-type inequality on Heisenberg groups.
The plan of the paper is as follows. In Section 2 we introduce a class of Greiner-type vector field and prove the corresponding weighted -Rellich inequality on H-type groups; Section 3 is devoted to the proof of weighted Hardy- and Rellich-type inequalities on nonisotropic Heisenberg groups and a Rellich-Sobolev-type inequality on Heisenberg groups.
2. Rellich Inequality on H-Type Groups
Note that when and becomes the sub-Laplacian on the H-type group If , and , is a Greiner operator (see [5, 20]). Also we note that vector fields are neither left nor right invariant and they do not satisfy Hörmander's condition for
The main results in  are the following.
Now we prove Theorem 2.4.
we have thus proved (2.15).
The following is an immediate consequence of Theorem 2.4, which is an extension of the uncertainty principle inequality.
We mention that when and our inequality (2.34) goes back to inequality (5.7) in  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 and Proposition in  and the weighted Hardy inequality (Theorem ) in .
3. Rellich Inequality on Nonisotropic Heisenberg Groups
For the proof of the above inequality, we need the following lemma which can be proved by a similar method in .
We now prove Theorem 3.1.
Now it is time to prove the following Rellich inequality on nonisotropic Heisenberg group.
We now give the proof of Theorem 3.3.
Combining with (3.22), (3.23), and the 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.
The following is also an uncertainty principle type inequality which is an immediate consequence of Theorem 3.3.
With the help of Theorem 3.6, we can also obtain a Rellich-Sobolev-type inequality on the Heisenberg group.
Thus we obtain the desired result.
The authors would like to thank Professor Genkai Zhang for constant encouragement and the referee for his/her helpful comments. The research was supported by the Natural Science Foundation of Zhejiang Province (no. Y6090359, Y6090383) and Department of Education of Zhejiang Province (no. Z200803357).
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