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

Let $v:[0,T]\times {\mathbb{R}}^d\to \mathbb{R}$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]{il} \partial{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t{)}_{t\in [0,T]}$ be the associated ${\mathbb{R}}^d$-valued diffusion process on some appropriate $(\mathrm{\Omega },\text{cF},\text{Q})$. For $p\in [2,\mathrm{\infty })$ and a measure $d\mathrm{\P }={\lambda }_{T}d\text{Q}$, where ${\lambda }_T$ satisfies the Muckenhoupt condition $A_{\alpha }$ for $\alpha \in (1,p)$,werelatethebehaviorof$|g(X_T)-\ept g(X_T) |{L_p(\P)}$,$|\nabla v(t,X_t) |{L_p(\P)}$and$|D^2 v(t,X_t) |_{L_p(\P)}$ to each other, where D^2v:=(\partial_{x_i \partial_{x_l}v)_{i,l} is the Hessian matrix.