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 $epsilon>0$, as the sum of a nilsequence $phi(n)$ and a sequence $omega(n)$ satisfying $lim_{Ntoinfty}frac{1}{N}sum_{n=1}^N |omega(n)|<epsilon$ and $lim_{Ntoinfty}frac{1}{|mathbb{P}cap [N]|}sum_{pin mathbb{P}cap [N]} |omega(p)|<epsilon$.