An abstract uncertainty principle with applications

Under Wigdersons’ framework and by sorting out the technical points in the recent works of Tang (J. Fourier Anal. Appl. 31 (2025)) and Dias-Luef-Prata (J. Math. Pures Appl. (9) 198 (2025)), we prove an abstract uncertainty principle for functions in the $L^p$ setting. An immediate consequence is a new uncertainty principle for the Fourier transform, unifying and extending many existing results. More applications are shown for PDEs, including the moment growth estimates for some linear and nonlinear dispersive equations, and a type of weighted lower bound estimate for the spacetime moment of the Schrödinger equation and heat equation inspired by the control theory.