On the number of prime factors with a given multiplicity over h-free and h-full numbers

Let $k$ and $n$ be natural numbers. Let ${\omega }_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of ${\omega }_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that ${\omega }_1(n)$ has normal order $\log \log n$ over $h$-free numbers, ${\omega }_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ${\omega }_k(n)$ with $1<k<h$ do not have normal order over $h$-free numbers and ${\omega }_k(n)$ with $k>h$ do not have normal order over $h$-full numbers.