A bounded property for gradients of diffusion semigroups on Euclidean spaces

We consider uniformly elliptic diffusion processes X(t,x) on Euclidean spaces Rd, with some conditions in terms of the drift term (see assumptions A2 and A3). By using interpolation theory, we show a bounded property which gives an estimate of ∇xE[ f(X(t,x))] involving |x| and || f||∞ but not ||∇f||∞, and a power of 1t smaller than 1.