I'm trying to prove that absolute integrability does not imply square integrability. I am able find a counterexample, for example $|x|^{-1/2} / (1+x^2)$, but I would like to find the conditions under which the implication does not hold.
I tried using Cauchy-Schwarz but that didn't get me anywhere. What would be a good starting point?
You will have to look into what is refered as $L_p$ norms (and Banach spaces), as these criteria involve different norms (e.g absolute value, square norm and so on). So the criteria are related to the norm used, especialy check Relations between p-norms