Distribution of $\omega (n)$ over $h$-free and $h$-full numbers

Let $\omega (n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $\omega (n)$ has normal order $\log \log n$ over naturals. In this work, we establish the first and the second moments of $\omega (n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $\omega (n)$ has normal order $\log \log n$ over these subsets.