Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$

Question

Let $\alpha \in (0, 1)$ and $\psi : \mathbb R^d \to \mathbb R^d$ be a $C^\infty$-diffeomorphism such that $\|\nabla \psi\|_\infty + \|\nabla \psi^{-1}\|_\infty < + \infty$. Let $$ I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp \left ( - \frac{|\psi(x) - y|^2}{t} \right) \, \mathrm d y. $$

If $\psi$ is the identity map, then (by a change of variable) there is a constant $C>0$ such that $$ I (t) \le C t^{\frac{\alpha}{2}}, \quad \forall t >0. $$

Is there such an upper bound for general $\psi$ where $C$ possibly depends on $\psi$?

Thank you so much for your elaboration?

Answer 1

$\newcommand\R{\mathbb R}\newcommand\al{\alpha}$No finite upper bound on $I(t)$ exists in such generality, even when $\psi(x)=2x$ for all $x$.

Indeed, using the substitution $y=\psi(x)-z\sqrt{t/2}$ and letting $Z$ denote the standard normal random vector in $\R^d$, we have $$I(t)=\sup_{x\in\R^d}I_x(t),$$ where $$I_x(t):=2^{-d/2}E\big|x-\psi(x)+Z\sqrt{t/2}\big|^\al \\ \ge2^{-d/2}(|x-\psi(x)|^\al-(t/2)^{\al/2} E|Z|^\al) \to\infty$$ as $|x|\to\infty$ if e.g. $\psi(x)=2x$.