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?
Asked
Active
Viewed 216 times
0
-
What does the notation inside the second expectation stand for? Is $D^n$ the $n^{th}$ derivative with respect to something? What is $H^k$ in the subscript? Last point: Why are you taking the expectation of what looks like a norm, which is another expectation? – Calculon Jul 22 '15 at 13:49
-
$H$ is a real separable Hilbert space. You can treat that $D^n$ to be the $n^{th}$ derivative. For example, $F=f(W(h_1)...W(h_n))$, then $DF=\sum \partial_i f(W(h_1)...W(h_n))h_i$. Note that W(h_{.}) is isonormal Gaussian process, $h$ lie in $H$. – will_cheuk Jul 23 '15 at 01:49
1 Answers
1
Hints:
- prove first that $\|F\|_{k,p} \leq \|F\|_{j,p}$, i.e. with the same $p$ but $k \leq j$.
- Use the fact that you are in a probability space and invoke the standard inclusion for $L^p$ spaces.
