What is the minimum sampling duration to capture a given frequency?
Using the DFT, the lowest bin is indeed corresponding to one period (i.e., the frequency which have exactly one period within the length of the DFT).

Conceptually, the DFT is correlating the input signal with different sinusoids. Now, to capture the phase information, these are "complex sinusoids", but if you knew the phase beforehand, you could just correlate with a real-valued sinusoid with the same phase to obtain the amplitude of the sinusoid.

Hence, it would conceptually be possible to register the existence of 20 Hz using a dozen sample at 44 kHz if you knew the phase of the signal (or by making multiple computations with different phase shifts). The drawback is that there are (probably/most likely) other signals which will be falsely detected as 20 Hz, because they correlate as well as the 20 Hz signal (and/or cancel the 20 Hz signal).

This is in no way a comprehensive answer, but more an insight that what the DFT and most other transforms do is to correlate the input signal to a set of basis functions, to see how they compare. With enough knowledge of your input signal you can find simpler basis functions for your particular case, but there is a trade-off between flexibility (ability to handle different cases) and simplicity (how little information/computation is needed).