Let $(\mathcal{F}_t)_{t\in [0,T]}$ be a filtration on a probability space $\Omega$. Fix $1<p<\infty$. Let $\{\mathcal{E}(\cdot|\mathcal{F}_t)\ :\ t\in [0,T] \}$ the associated family of conditional expectation operators on $L^p(\Omega)$. Fix $f\in L^p(\Omega)$. Consider the map $$ t\mapsto \mathcal{E}(\cdot|\mathcal{F}_t)f. $$
I am interesting in information on the derivative of this map (possibly with stronger assumptions). Do you know references or results in this area?
remark: I know nothing information on the existence of this derivative.
