Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$.
It is not hard to show that $$ \lVert \mathbb{E}[X a(X)] \rVert_2^2 \lesssim \mathbb{E}[a(X)]^2 \log \frac{1}{\mathbb{E}[a(X)]} \tag{1} $$ e.g., via Gibbs variational principle.
Does (1) have an analogue for more general functions, say convex $\phi\colon \mathbb{R}^d \to \mathbb{R}$ ($\phi(X)$ instead of $X$)? That is, to bound $\mathbb{E}[\phi(X) a(X)]$?
I am mostly interested in $\phi(x) = e^{\delta \lVert x\rVert_2^2}$, where $\delta>0$ is an arbitrary (small enough) parameter.
Is there a way to derive non-trivial (i.e., non-Cauchy-Schwarz-y) upper bounds on $\mathbb{E}[e^{\delta \lVert X\rVert_2^2} a(X)]$, of the form $$ \mathbb{E}[e^{\delta \lVert X\rVert_2^2} a(X)] \leq \Psi(\mathbb{E}[a(X)]) \tag{2} $$where $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$?
Note that Cauchy—Schwarz, ignoring the Gaussianity, would give $$\Psi(\mathbb{E}[a(X)]) = \sqrt{\mathbb{E}[a(X)]}\cdot{(1-4\delta)^{-d/4}} \tag{3}$$ while in view of (1) one may hope a nearly-linear dependence on $\mathbb{E}[a(X)]$.
