Norms of the Lattice Point Discrepancy

We estimate the [Formula omitted] norms of the discrepancy between the volume and the number of integer points in [Formula omitted], a dilated by a factor r and translated by a vector x of a convex body [Formula omitted] in [Formula omitted] with smooth boundary with strictly positive curvature, [integral]R[integral]Td[summation]k[element of]Zd[chi]r[Omega]-x(k)-rd[Omega]pdxd[mu](r-R)1/p,where [Formula omitted] is a Borel measure compactly supported on the positive real axis and [Formula omitted].