Mapping Properties of Maximal Functions on Graded Lie Groups

This survey reviews the works [6, 7–8], on maximal operators on graded Lie groups. In the first work [8], in collaboration with M. Ruzhansky and J. Delgado, we extend the classical theorem by Calderón, Coifman, and Weiss on the dyadic maximal function in Rⁿ to graded Lie groups, providing its boundedness in terms of criteria involving the group Fourier transform. In the second article [6], we generalize Bourgain’s weak-type nn-1,1 inequality for spherical averages, using Rockland operators in this context. Finally, the third work [7] formulates an extension of Bourgain’s result for maximal functions associated with kernels having differentiable Fourier transforms, providing sharp L²-norm estimates. These results are discussed in this survey emphasising the importance of the Fourier analysis on graded Lie groups and of Rockland operators in the analysis of maximal operators on graded Lie groups. These articles formed the foundation of a summer course delivered at the Yerevan State University, Armenia, as part of a Lecture Course at the Summer School “Analysis, PDEs and Applications GMG’70”, dedicated to the 70th anniversary of Prof. Martin Grigoryan, held from June 24 to July 6, 2024.