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Uncertainty principles and characterization of the heat kernel for certain differential-reflection operators

Abstract We prove various versions of uncertainty principles for a certain Fourier transform ℱA. Here, A is a Chébli function (that is, a Sturm–Liouville function with additional hypotheses). We mainly establish an analogue of Beurling's theorem, and its relatives such as theorems of Gelfand–Shilov type, of Morgan type, of Hardy type, and of Cowling–Price type, for ℱA and relate them to the characterization of the heat kernel corresponding to ℱA. Heisenberg's and local uncertainty inequalities are also proved.