Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > 0$. For example, i have in mind functions of the form $m(x) = x^\top A x$, for some $d \times d$ matrix $A$.
Question 1. Under what conditions on $m$ is there an integration by parts formula (i.e a Stein's lemma) of the form $$ E[X_1 g(\sum_i u_i X_i)] = u_1 E[g'(\sum_i u_i X_i)], $$ for any smooth function $g:\mathbb R \to \mathbb R$ and vector $u=(u_1,\ldots,u_1)\in \mathbb R^d$.
In particular,
Question 2. Is the condition that $m$ is symmetric in the sense that $m(-x)=m(x)$ for all $x$ sufficient ? If not, what about the condition $m(x) = x^\top A x$ for some matrix $A$ (e.g rank 1, i.e $A=uv^\top$) ?
