Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

© 2016 Elsevier Inc. We extend the recent L1 uncertainty inequalities obtained in [13] to the metric setting. For this purpose we introduce a new class of weights, named isoperimetric weights, for which the growth of the measure of their level sets μ can be controlled by rI(r), where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new localized Poincaré inequalities, and interpolation, to prove Lp, 1≤p<∞, uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities of [20], and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.