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Sharp estimates for Hardy operators on Heisenberg group
Qingyan WU,Zunwei FU
Front. Math. China. 2016, 11 (1): 155-172.
https://doi.org/10.1007/s11464-015-0508-5
In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on Lp(Hn) is still p/(p−1). This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝn, or ‘ellipsoids’ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.
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