Mixed Lp (L2) norms of the lattice point discrepancy

We estimate some mixed Lp(L2) norms of the discrepancy between the volume and the number of integer points in rΩ − x, a dilation by a factor r and a translation by a vector x of a convex body Ω in R^d with smooth boundary with nonvanishing Gaussian curvature, (int_{T^d} ( (1/H) int_R^{R+H} |sum_{kin Z^d} χ_{rΩ−x}(k)−r^d |Ω| dr)^{p/2} dx)^{1/p}. We obtain estimates for fixed values of H and R → ∞, and also asymptotic estimates when H → ∞.