Complete higher-order difference characterization

Establish a complete characterization of Besov and Triebel--Lizorkin spaces on general filtered Lie groups by higher-order differences, including the reverse inequality that is not obtained by the techniques used in the paper.

Background

The paper proves first-order difference characterizations for Besov and Triebel--Lizorkin spaces on filtered Lie groups and obtains some one-sided estimates for higher-order differences. It also proves the reverse inequality for second-order differences in the Besov case, but not a full higher-order result.

The unresolved difficulty is the absence, for general filtered Lie groups, of an analogue of the Fourier-analytic inversion argument used for higher-order differences in Euclidean settings. The authors explain that transferring higher-order difference characterizations from the homogeneous-group contraction to the original filtered Lie group is obstructed by the lack of sufficient comparability between their local geometries.