Let be a measure space. Use Hölder's inequality to prove that if and then for every , , and with .
I don't see how to prove this right now. I know that this looks a lot like Young's inequality which follows from Hölder's inequality, so I was thinking to adopt a similar proof of that theorem. Right now, I just need a few hints on how to proceed. Thank you.
Simply apply Hölder's inequality with the correct p and q... Write down what you want to prove ( ), rewrite it like Holder's inequality ( ) in order to guess the exponents, and its shouldn't be a problem.