Lp Norms of the Lattice Point Discrepancy

We estimate the L^p norms of the discrepancy between the volume and the number of integer points in rO− x, a dilated by a factor r and translated by a vector x of a convex body O in R^d with smooth boundary with strictly positive curvature, (int_Rint_{T^d}|sum_{kin Z^d} chi_{rO-x}(k)−r^d|O||^p dxdμ(r−R))^{1/p} , where μ is a Borel measure compactly supported on the positive real axis and R →+∞.