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Sharp estimates for Hardy operators on Heisenberg group

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Abstract

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 L p(ℍn) is still p/(p-1). This goes some way to imply that the L p 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 H 1 to L 1. Moreover, we describe the difference between M p weights and A p weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

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Authors and Affiliations

  1. Department of Mathematics, Linyi University, Linyi, 276005, China

    Qingyan Wu & Zunwei Fu

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  1. Qingyan Wu
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  2. Zunwei Fu
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Correspondence to Zunwei Fu.

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Wu, Q., Fu, Z. Sharp estimates for Hardy operators on Heisenberg group. Front. Math. China 11, 155–172 (2016). https://doi.org/10.1007/s11464-015-0508-5

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  • DOI: https://doi.org/10.1007/s11464-015-0508-5

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