Fractional smoothness of functionals of diffusion processes under a change of measure

Let v: [O, T] × R^d → R be the solution of the parabolic backward equation ∂_tv + (1/2) Σ _i,j[δ δ^T]_i,j ∂_x,i ∂_x,j v + Σ_i b_i ∂ _x,i v + kv = 0 with terminal condition g, where the coefficients are time-and state-dependent, and satisfy certain regularity assumptions. Let X = (X_t)_t∈[0,T] be the associated R^d-valued diffusion process on some appropriate (Ω, F, Q). For p ∈ [2, ∞) and a measure dP = λT dQ, where λT satisfies the Muckenhoupt condition Ap, we relate the behavior of ∥g(XT) - EP(g(XT)|Ft)∥Lp (P), ∥∇v(t, _Xt)∥Lp (P), ∥D²v(t, Xt)∥Lp (P) to each other, where D²v:= (∂xi ∂xj ^v)i,j is the Hessian matrix.