Question: Let belong to both and , with . Show that for all .
There is a pretty neat trick to do this question, known as the “interpolation technique”. The proof is as follows.
For , there exists such that . This is the key “interpolation step”. Once we have this, everything flows smoothly with the help of Holder’s inequality.
Note that the magical thing about the interpolation technique is that and are Holder conjugates, since is easily verified.