The Hardy--Littlewood maximal operator plays a very important role in harmonic analysis and partial differential equations. In this talk, we introduce a new family of `lifted' maximal operators, called the lifted Hardy—Littlewood (LHL) maximal operators. We establish their sharp Lp estimates for any p∈[1,∞) and present their applications to establish the weak-type characterization of the Lp norm of various important differential operators, which generalizes the recent surprising formula on the gradient of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung. On the other hand, we also discuss the application of LHL maximal operators to the weak-type characterizations of Hardy(--Sobolev) spaces in terms of truncated Riesz transforms.
