Hardy-type inequalities on a half-space in the Heisenberg group
© Liu and Luan; licensee Springer 2013
Received: 30 April 2012
Accepted: 23 May 2013
Published: 10 June 2013
We prove some Hardy-type inequalities on half-spaces for Kohn’s sub-Laplacian in the Heisenberg group. Furthermore, the constants we obtained are sharp.
where , and the constant is sharp. This shows that the Hardy constant jumps from to when the singularity of the potential reaches the boundary. For more information about this inequality and its applications, we refer to [3–10] and the references therein.
where is the horizontal gradient associated with Kohn’s sub-Laplacian on (for details, see Section 2). Furthermore, the constant in (1.3) is sharp (see , Theorem 3.13). In this note we shall show that when the singularity is on the boundary, the Hardy constant also jumps. In fact, we have the following.
where , and the constant in (1.4) is sharp.
With the same technique, we obtain the following sharp Hardy inequality on .
Furthermore, the constant in (1.5) is sharp.
To get the last equation, we use the fact .
Here we use the sharp Hardy inequality (1.2). This completes the proof of Theorem 1.1. □
The proof of Theorem 1.2 is therefore completed. □
The first author is supported by the National Natural Science Foundation of China (No. 11171259) and the second author is supported by the National Natural Science Foundation of China (No. 11201346).
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