Pointwise convergence of polynomial multiple ergodic averages along the primes
We establish pointwise almost everywhere convergence for the polynomial multilinear ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T \colon X \to X$ is an invertible measure-preserving transformation of a probability space $(X,\nu)$, $P_1,\ldots, P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\ldots,f_k\in L^\infty(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. We develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages in the general $k$-linear setting. The advantage of our method, besides establishing Weyl-type inequalities for multilinear Cram{\'e}r-weighted averages and sharp $p$-adic $L^q$-improving multilinear estimates among other tools, is that for the first time it allows us to work with inverse theorems having subpolynomial bounds in the general multilinear setting. This, in turn, yields sharp $r$-variational estimates $r > 2$ for our weighted polynomial multilinear ergodic average and, more importantly, offers prospects for addressing other multilinear problems involving inverse theorems lacking polynomial bounds.