Hölder's inequality. Understanding proof?!

Question

I know how most standard textbooks show that $||f*g||_r \le ||f||_p||g||_q$ with $\frac{1}{r}+1=\frac{1}{p}+ \frac{1}{q}$, but I found a book where the hint $|f(x-y)g(y)|\le (|f(x-y)|^p|g(y)|^q)^{\frac{1}{r}}|f(x-y)|^{1-\frac{p}{r}}|g(y)|^{1-\frac{q}{r}}$ is given and it is said that one just needs to apply Hölder's inequality. But honestly, I don't see it how this follows. To which coefficients is Hölder's inequality applied here? Just in order to clarify this. It is clear how this expansion happened. All I am asking is how you get from the 2nd line to the first one by applying Hölder's inequality?

Answer 1

Sorry I misunderstood your question first. However, your proof is given here: