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 $r\mathrm{\Omega }-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $\mathrm{\Omega }$ in ${\mathbb{R}}^d$, $ \left{ {\int_{$\mathbb{T}$^{d}}}\left( \frac{1}{H} {\int_{R}^{R+H}}\left\vert \sum_{k\in$\mathbb{Z}$^{d}}\chi _{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert \right\vert^{2}dr\right)^{p/2}dx\right} ^{1/p}. $ We obtain estimates for fixed values of $H$ and $R\to \mathrm{\infty }$, and also asymptotic estimates when $H\to \mathrm{\infty }$.