A well known useful inequality of Gallagher states (in one form) that for any sequence $a:\mathbb N\to\mathbb C$, we have that $$\int_{|\theta|\le\delta} \bigg|\sum_n a(n)e(n\theta)\bigg|^2d\theta\ll \int_{-\infty}^\infty \bigg|\delta\sum_{x\le n\le x + (2\delta)^{-1}}a(n)\bigg|^2dx.$$ Is there any analogous tool which allows one to bound an average over sums over short intervals by an integral of the exponential sum over a short arc (perhaps with some restrictions on the sizes of the coefficients and ranges of the parameters)?
$\begingroup$
$\endgroup$
Add a comment
|