A decomposition of multicorrelation sequences for commuting transformations along primes

We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $ϵ>0$, as the sum of a nilsequence $\varphi (n)$ and a sequence $\omega (n)$ satisfying ${\lim }_{N\to \mathrm{\infty }}\frac{1}{N}{\sum }_{n=1}^N\mathrm{\mid }\omega (n)\mathrm{\mid }<ϵ$ and $\lim_{N\to\infty}\frac{1}{|$\mathbb{P}$\cap [N]|}\sum_{p\in $\mathbb{P}$\cap [N]} |\omega(p)|<\epsilon$.