An improved Hardy type inequality on Heisenberg group
© Xiao; licensee Springer. 2011
Received: 2 April 2011
Accepted: 25 August 2011
Published: 25 August 2011
Motivated by the work of Ghoussoub and Moradifam, we prove some improved Hardy inequalities on the Heisenberg group via Bessel function.
Mathematics Subject Classification (2000):
KeywordsHardy inequality Heisenberg group
has a positive solution on (0, ρ]. These include inequalities (1.2)-(1.4).
To prove the main result, we first need the following preliminary result.
where r = |x| andis the radial derivation.
Observe that if f is radial, i.e., f(x) = f(r), then |∇ f| = |∂ r f|. By inequality (1.5), inequality (2.1) holds.
This completes the proof of lemma 2.1.
Proof of Theorem 1.1
Here we use the fact ⟨x, Λx⟩ = 0 since Λ is a skew symmetric matrix.
The proof of Theorem 1.1 is completed.
Proof of Corollary 1.2
Here we use the fact that the sharp constant in inequality (1.3) is 1/4 (see ). This completes the proof of Corollary 1.2.
Proof of Corollary 1.3
This completes the proof of Corollary 1.3.
The authors thanks the referee for his/her careful reading and very useful comments which improved the final version of this paper.
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