when looking at the amplitude envelope of an oscillation (see red trace in Figure below), it seems to me as if the frequency content of the envelope depends on the the frequency of the underlying carrier signal (blue trace). More specifically, it looks as if the fastest fluctuation the envelope can exhibit is related to the frequency of the carrier signal. There should be a straightforward mathematical relation between the carrier frequency and the fastest frequency in the envelope. If I think about the individual peaks of the oscillation as "samples" for the envelope, then the fastest frequency the envelope contains should be carrier_freq/2, but I'm not sure if this is correct. I'd appreciate any help or hints in the right direction.
Your intution is right, the carrier frequency has to be at least twice the highest frequency in your signal. Practically, much higher carrier frequencies are chosen, as you would need ideal or nearly ideal lowpass filters for carrier frequencies this low.