Mixed L^{p}(L^{2}) norms of the lattice point discrepancy

We estimate some mixed L^{p}\left ( L^{2}\right ) norms of the discrepancy between the volume and the number of integer points in r\Omega -x, a dilation by a factor r and a translation by a vector x of a convex body \Omega in \mathbb{R}^{d} with smooth boundary with nonvanishing Gaussian curvature, \displaystyle \left \{ {\displaystyle \int _{\mathbb{T}^{d}}}\left ( \dfrac {1}... ...vert \Omega \right \vert \right \vert ^{2}dr\right ) ^{p/2}dx\right \} ^{1/p}. We obtain estimates for fixed values of H and R\to \infty , and also asymptotic estimates when H\to \infty .