Weighted uniform distribution of subpolynomial functions along primes and applications

Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n){)}_{n\in \mathbb{N}}$ and $(u(p_n){)}_{n\in \mathbb{N}}$, where $p_n$ denotes the $n$-th prime. This extends the main result of [4] to the weighted setting and leads to new applications in uniform distribution theory, ergodic theory, and additive combinatorics.