Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^n$ be random variables with characteristic functions $\phi_X(t)$ and $\phi_Y(t)$, respectively.

Suppose that \begin{align} \sup_{t \in \mathbb{R}^n} \frac{|\phi_X(t)-\phi_Y(t)|}{\|t\|} \le \epsilon. \quad (*) \end{align}

Question: Can we say something about how close the distributions of $X$ and $Y$ are in some metric over probability spaces? In other words, suppose that $(*)$ small what other distance would be small too.

For example, for $n=1$. There exists the following inequality \begin{align} L^2(P_X,P_Y) \le 2 \sup_{t \ge 0} \frac{|\phi_X(t)-\phi_Y(t)|}{t} \end{align} where $L$ is the L\'evy distance.

However, I was not able to locate similar inequality for $n>1$. One distance that I have taken a look at extensively is the Levy-Prokhorov metric. However, whenever I find anything it always depends on the derivatives of characteristic functions.