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）. ...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.展开更多
We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theo...We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.展开更多
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271175, 11301248, 11401287）, the Natural Science Foundation of Shandong Province （Grant No. ZR2012AQ026）, and the AMEP of Linyi University.
文摘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.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.12171221,12071197)the Natural Science Foundation of Shandong Province(Grant Nos.ZR2019YQ04,2020KJI002).
文摘We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type.In addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space.Finally,a concrete example shows that the local version of commutators also has an independent interest.
