Improving and Maximal Inequalities for Primes in Progressions

Assume that $y < N$ are integers, and that $(b,y) =1$. Define an average along the primes in a progression of diameter $y$, given by integer $(b,y)=1$. \begin{align*} A_{N,y,b} := \frac{\phi (y)}{N} \sum _{\substack{n <N\\n\equiv b\pmod{y}}} \Lambda (n) f(x-n) \end{align*} Above, $\Lambda$ is the von Mangoldt function and $\phi$ is the totient function. We establish improving and maximal inequalities for these averages. These bounds are uniform in the choice of progression. For instance, for $1< r < \infty$ there is an integer $N _{y, r}$ so that \begin{align*} \lVert \sup _{N>N _{y,r}} \lvert A_{N,y,b} f \rvert \rVert_{r}\ll \lVert f\rVert_{r}. \end{align*} The implied constant is only a function of $r$. The uniformity over progressions imposes several novel elements on the proof.