On the smoothness of conditional expectation functionals

Given a class ΛΛ of real functions and a twice differentiable real-valued map φφ on R, let ΓΓ be an R-valued functional on ΛΛ of form Γ:λ↦E[Z⋅φ(E[Y∣λ(X)])], where ZZ and YY are random variables and XX is a random vector. This paper calls ΓΓ a conditional expectation functional  . Conditional expectation functionals often arise in semiparametric models. The main contribution of this paper is that it provides nontrivial conditions under which ΓΓ has a uniform modulus of continuity with order 2. Hence under these conditions, the functional ΓΓ becomes very smooth.