Uncertainty Principles for the Ambiguity Function

We extend uncertainty principles which are valid for the Fouriertransform to the setting of the ambiguity function. A generalresult is established for annihilating sets: strongly/weaklyannihilating sets for the Fourier transform yield such setsfor the ambiguity function, extending a result known for setsof finite measure. We apply this to sublevel sets of nondegeneratequadratic forms. Our main result is a sharp version of Beurling'suncertainty principle for the ambiguity function.