Let $p$ and $q$ be probability distributions on a metric space $X=(X, d)$ with densities $dp$ and $dq$, such that there exists $0 < \alpha < \beta < \infty$ satisfying
$$
\alpha d p \le dq \le \beta dp .
$$
What is an upper bound for the Wasserstein distance $W_d(p,q)$ ?
Notes: $W_d(p, q) := \sup_{\|f\|_{\text{Lip}} \le 1} |\mathbb E_{x \sim p}[f(x)] – \mathbb E_{x \sim q}[f(x)]|$
Update
It's well known (e.g see Gabe K's response below or Theorem 4 of this paper) that
$$
W_d(p, q) \le \operatorname{diam}(X)\operatorname{TV}(p, q).
$$
Thus if $X$ has finite diameter, it suffices to bound $\operatorname{TV}(p, q)$.
Recall the definition of total variation,
$$
\operatorname{TV}(p, q) := \sup_{A \subseteq X} \left|\int_A dq-\int_A dp\right|.
$$
Now, for any $A \subseteq X$, one has
$
\int_A dq – \int_A dp \ge \int_A(\alpha-1)dp = (\alpha-1)p(A).
$
Similarly, one has $\int_A dq – \int_A dp \le (\beta-1)p(A)$. Thus
$$
\left|\int_A dq – \int_A dp \le (\beta-1)p(A)\right| \le \max(1-\alpha,\beta-1)p(A) \le \max(1-\alpha,\beta-1),
$$
and so $\operatorname{TV}(p,q) \le \max(1-\alpha,\beta-1)$. Putting things together, we get
$$
W_d(p,q) \le \operatorname{diam}(X)\max(1-\alpha,\beta-1).
$$
Case of infinite diameter
In case of infinite diameter assuming that $p$ or $q$ satisfies a Talagrand transportation-cost inequality, i.e
$$W_{d,2}(p,r)\le \sqrt{(2/\rho)\operatorname{kl}(r\|p)}
$$
for some $\rho > 0$ and for any distribution $r$ on $X$ relatively continuous w.r.t $p$,
with a similar story for $q$ (e.g for standard Gaussian, this holds with $\rho=1$), allows us to still get an upper bound, namely
$$
W_{d,2}(p,q) \le \sqrt{2\min (\beta\log(\beta)/\rho_p,(1/\alpha)\log(1/\alpha)/\rho_q)}.
$$
Best Answer
Any estimate will need to incorporate the diameter of $X$. To see this, consider the interval $[0,L]$ and the distribution with density,
$$\begin{cases} \frac{2}{L} & 0 \leq x \leq L/3 \\ \frac{1}{2L} & L/3 \leq x \leq L \\ \end{cases} $$
One can calculate the distance from the uniform distribution $1/L$ using the dual formulation of the Wasserstein distance, and it blows up as $L$ goes to infinity. Therefore, we cannot hope for an estimate solely in terms of $\alpha$ and $\beta$.
However, assuming a bound on the diameter $D$ of $X$, then we can say more. With $\alpha$ and $\beta$, we can derive a bound on $\delta(p,q)$ the total variation distance between $p$ and $q$. One possible transfer plan from $p$ to $q$ is to move the excess mass of $p$ to $q$, fixing the overlap. The total mass to be moved then is $\delta(p,q)$, and the cost to move the mass is no more than $D^d$. Skipping some derivations, this shows that the Wasserstein 2-distance is bounded by $D \sqrt{ \delta(p,q)}$, so this gives your desired result. This can be generalized to the other Wasserstein metrics as well.