I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem.

Let $B^n$ be $n$-dimensional balls and $h:B^{n-k}\times B^{k} \to U$ a $k$-dimensional foliation chart with $C^\infty$ leaves. i.e. For $x\in B^{n-k}$, $W_x := h(\{x\}\times B^k)\subset U$ are $C^\infty$ submanifolds. Furthermore let $y\in B^{n-k}$ such that $L=h(\{y\}\times B^{n-k})$ is a smooth transversal. Then according to Brin-Stuck "Introduction to Dynamical Systems" the foliation is *absolutely continuous* if there is a measurable family of positive measurable functions $\delta_x: W_x\to\mathbb R$ such that for all measurable sets $A\subset U$
$$\int_U \mathbf 1_A(x,y) dm_{euc} = \int_L \int_{W_x} \mathbf 1_A(x,y) \delta_x(y) dm_{W_x}(y)dm_L(x).$$
Here $dm_{euc}$ is the Eucildean measure on $U$ and $dm_{W_x}$ and $dm_L$ the measures obtained by restricting $dm_{euc}$ on the smooth submanifolds.

If the foliation ist the stable foliation of an Anosov differomorphism then I found that it is absolutely continuous. It think it is clear that one cannot hope that $\delta_x(y)$ depends $C^\infty$ on $x$ unless the foliation is smooth.

**My question is**: Under which conditions on the foliation can one obtain, that for any $x$ one has $\delta_x \in C^\infty(W_x)$(ideally I would also like to have, that $\delta_x$ depends continuousely on $x$ w.r.t. to the $C^\infty$-topology, but maybe I can also handle the case if it is only measureable)? And more precisely: Is such a statement always true for the strong stable foliation of a $C^\infty$ Anosov flow. If it doesn't hold for general Anosov Flows: What are the weakest known assumption that it holds? (If I understand it correctly then $C^1$ foliations implies this statement via Fubinis theorem).