Measure change bound for function of subgaussian r.v

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 $$\mathbb{E}[a(X)]$$.

• Do you have more information about the function $a$?
– user114668
Feb 13 '19 at 20:24
• @student no, npt really. Feb 13 '19 at 20:27