Christina Giannitsi, Georgia Institute of Technology
Authors: Christina Giannitsi, Michael Lacey, Hamed Mousavi and Yaghoub Rahimi
2022 AWM Research Symposium
Women in Analysis Research Network - Special Session for Graduate Students and Postdoctoral Fellows
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 $b$. $$A_{N,y,b} := \frac{\phi (y)}{N} \sum_{\substack{n < N\\n\equiv b\mod y}} \Lambda (n) f(x-n),$$ where $\Lambda$ is the von Mangoldt function and $\phi$ is the totient function. We establish improving and maximal inequalities for these averages, with bounds that are uniform in the choice of progression. For instance, for $ 1< r < \infty $ there is an integer $N _{y, r}$ so that $$\|\sup _{N>N _{y,r}} | A_{N,y,b} f | \| _{r} \ll \| f \| _{r}.$$ The implied constant is only a function of $r$.
This is joint work with Michael Lacey, Hamed Mousavi and Yaghoub Rahimi.