How to apply Holder inequality to prove the following?

Question

In the book "Malliavin Calculus and related topics", the author states that $||F||_{k,p}=((E(|F|^p)+\sum_{n=1}^k E(||D^n F||^p_{H^k}))^{\frac{1}{p}}$ has monotonicity property, i.e. $||F||_{k,p}\leq ||F||_{j,q}$ when $k\leq j$ and $p\leq q$ if $F$ is smooth random variable. To complete the proof, I need to know how to prove for the case when $p<q$ and $k=j$. Someone suggests to use Holder inequality, but in general, $q$ and $p$ are not conjugate. How to apply Holder inequality to prove it? If Holder inequality cannot be used to prove it, how to arrive the conclusion then?

Answer 1

Hints:

1. prove first that $\|F\|_{k,p} \leq \|F\|_{j,p}$, i.e. with the same $p$ but $k \leq j$.
2. Use the fact that you are in a probability space and invoke the standard inclusion for $L^p$ spaces.