Sharp estimates for Hardy operators on Heisenberg group

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 H¹ to L¹. 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.

Keywords

Heisenberg group / Hardy operator / M_p weight

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