Divisors of an Integer in a Short Interval

Let $\mathcal{D}_{n} \subset \mathbb{N}$ denote the set of the $\tau(n)$ divisors of $n$. We study the function $$D_{n}(X,Y):=|\{d \in \mathcal{D}_{n}:\ X \le d \le X+Y\}|$$ for $Y \le X$.