Denote by $P\_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by endowing it with the $\sigma$-algebra generated by open sets and Wiener-measure. 


Let $F:P\_0(\mathbb{R}^d)\rightarrow \mathbb{R}$ be a cylinder function, i.e., $F(\omega)=f(\omega\_{s\_1},...,\omega\_{s\_n})$, where $0<s\_1<...<s\_n<1$ and $f$ and all its derivatives have at most polynomial growth. 


How can I see that the limit for a Cameron Martin path $h\in\mathcal{H}$


$\lim\_{t\rightarrow 0} \frac{F(\omega+th)-F(\omega)}{t}$ exists in at least $L^1$?


Or put differently, why can I interchange the limit with the integral sign in 


$\lim\_{t\rightarrow 0}\int\_{P\_0(\mathbb{R}^d)}\frac{F(\omega+th)-F(\omega)}{t}d\mathbb{P}(\omega)$?


**APPROACH**


In order to interchange the limit with the integral it would suffice to find an integrable function $G$ such that $F'(\omega+th)\leq G(\omega)$ for $t\in (-\epsilon, \epsilon)$. But I only see that 


$F'(\omega+th)=\sum\_{j=1}^n \nabla^jf(\omega\_{s\_1}+th\_{s\_1},...,\omega\_{s\_l}+th\_{s\_l})\cdot h\_{s\_j} 
$ $\leq\sup\_{t\in (-\epsilon,\epsilon)} \sum\_{j=1}^n\nabla^jf(\omega\_{s\_1}+th\_{s\_1},...,\omega\_{s\_l}+th\_{s\_l})\cdot h\_{s\_j}$


But how do I see that the last line in integrable? I understand that without the sup this line in integrable as $f$ is polynomial and $B\_{s\_i}$ has moments of all orders but this does not seem to suffice here.



Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?
We can assume that all the first derivatives of $f$ are bounded by some polynomial function, let's say $|\nabla^j f(x)|\le C(1+|x|^p)$. Then we have $|F'(\omega + th)| \le n C(1 + (\|\omega\|\_\infty + t \|h\|\_\infty)^p)$ which can be bounded, independent of $t$, by something like $C\_1(1 + \|\omega\|\_\infty^p)$. But by Fernique's or Skorohod's theorem, $\|B\|\_\infty$ has finite moments of all orders, i.e. $\int \|\omega\|\_\infty^p \,d\mathbb{P}(\omega) < \infty$.


